TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
I:
GENERALITIES
Shinichi
Mochizuki
March
2012
Abstract.
This
paper
forms
the
first
part
of
a
three-part
series
in
which
we
treat
various
topics
in
absolute
anabelian
geometry
from
the
point
of
view
of
develop-
ing
abstract
algorithms,
or
“software”,
that
may
be
applied
to
abstract
profinite
groups
that
“just
happen”
to
arise
as
[quotients
of]
étale
fundamental
groups
from
algebraic
geometry.
One
central
theme
of
the
present
paper
is
the
issue
of
under-
standing
the
gap
between
relative,
“semi-absolute”,
and
absolute
anabelian
geometry.
We
begin
by
studying
various
abstract
combinatorial
properties
of
profi-
nite
groups
that
typically
arise
as
absolute
Galois
groups
or
arithmetic/geometric
fundamental
groups
in
the
anabelian
geometry
of
quite
general
varieties
in
arbitrary
dimension
over
number
fields,
mixed-characteristic
local
fields,
or
finite
fields.
These
considerations,
combined
with
the
classical
theory
of
Albanese
varieties,
allow
us
to
derive
an
absolute
anabelian
algorithm
for
constructing
the
quotient
of
an
arithmetic
fundamental
group
determined
by
the
absolute
Galois
group
of
the
base
field
in
the
case
of
quite
general
varieties
of
arbitrary
dimension.
Next,
we
take
a
more
detailed
look
at
certain
p-adic
Hodge-theoretic
aspects
of
the
absolute
Galois
groups
of
mixed-characteristic
local
fields.
This
allows
us,
for
instance,
to
derive,
from
a
certain
result
communicated
orally
to
the
author
by
A.
Tamagawa,
a
“semi-absolute”
Hom-version
—
whose
absolute
analogue
is
false!
—
of
the
an-
abelian
conjecture
for
hyperbolic
curves
over
mixed-characteristic
local
fields.
Finally,
we
generalize
to
the
case
of
varieties
of
arbitrary
dimension
over
arbitrary
sub-p-adic
fields
certain
techniques
developed
by
the
author
in
previous
papers
over
mixed-characteristic
local
fields
for
applying
relative
anabelian
results
to
obtain
“semi-absolute”
group-theoretic
contructions
of
the
étale
fundamental
group
of
one
hyperbolic
curve
from
the
étale
fundamental
group
of
another
closely
related
hyperbolic
curve.
Contents:
§0.
Notations
and
Conventions
§1.
Some
Profinite
Group
Theory
§2.
Semi-absolute
Anabelian
Geometry
§3.
Absolute
Open
Homomorphisms
of
Local
Galois
Groups
§4.
Chains
of
Elementary
Operations
Appendix:
The
Theory
of
Albanese
Varieties
2000
Mathematical
Subject
Classification.
Primary
14H30;
Secondary
14H25.
Typeset
by
AMS-TEX
1
2
SHINICHI
MOCHIZUKI
Introduction
The
present
paper
is
the
first
in
a
series
of
three
papers,
in
which
we
con-
tinue
our
study
of
absolute
anabelian
geometry
in
the
style
of
the
following
papers:
[Mzk6],
[Mzk7],
[Mzk8],
[Mzk9],
[Mzk10],
[Mzk11],
[Mzk13].
If
X
is
a
[geometrically
def
integral]
variety
over
a
field
k,
and
Π
X
=
π
1
(X)
is
the
étale
fundamental
group
of
X
[for
some
choice
of
basepoint],
then
roughly
speaking,
“anabelian
geometry”
may
be
summarized
as
the
study
of
the
extent
to
which
properties
of
X
—
such
as,
for
instance,
the
isomorphism
class
of
X
—
may
be
“recovered”
from
[various
quotients
of]
the
profinite
group
Π
X
.
One
form
of
anabelian
geometry
is
“relative
anabelian
geometry”
[cf.,
e.g.,
[Mzk3]],
in
which
instead
of
starting
from
[vari-
ous
quotients
of]
the
profinite
group
Π
X
,
one
starts
from
the
profinite
group
Π
X
equipped
with
the
natural
augmentation
Π
X
G
k
to
the
absolute
Galois
group
of
k.
By
contrast,
“absolute
anabelian
geometry”
refers
to
the
study
of
proper-
ties
of
X
as
reflected
solely
in
the
profinite
group
Π
X
.
Moreover,
one
may
consider
various
“intermediate
variants”
between
relative
and
absolute
anabelian
geometry
such
as,
for
instance,
“semi-absolute
anabelian
geometry”,
which
refers
to
the
situation
in
which
one
starts
from
the
profinite
group
Π
X
equipped
with
the
kernel
of
the
natural
augmentation
Π
X
G
k
.
The
new
point
of
view
that
underlies
the
various
“topics
in
absolute
anabelian
geometry”
treated
in
the
present
three-part
series
may
be
summarized
as
follows.
In
the
past,
research
in
anabelian
geometry
typically
centered
around
the
establish-
ment
of
“fully
faithfulness”
results
—
i.e.,
“Grothendieck
Conjecture-type”
results
—
concerning
some
sort
of
“fundamental
group
functor
X
→
Π
X
”
from
varieties
to
profinite
groups.
In
particular,
the
term
“group-theoretic”
was
typically
used
to
refer
to
properties
preserved,
for
instance,
by
some
isomorphism
of
profinite
∼
groups
Π
X
→
Π
Y
[i.e.,
between
the
étale
fundamental
groups
of
varieties
X,
Y
].
By
contrast:
In
the
present
series,
the
focus
of
our
attention
is
on
the
development
of
“algorithms”
—
i.e.,
“software”
—
which
are
“group-theoretic”
in
the
sense
that
they
are
phrased
in
language
that
only
depends
on
the
structure
of
the
input
data
as
[for
instance]
a
profinite
group.
Here,
the
“input
data”
is
a
profinite
group
that
“just
happens
to
arise”
from
scheme
theory
as
an
étale
fundamental
group,
but
which
is
only
of
concern
to
us
in
its
capacity
as
an
abstract
profinite
group.
That
is
to
say,
the
algorithms
in
question
allow
one
to
construct
various
objects
reminis-
cent
of
objects
that
arise
in
scheme
theory,
but
the
issue
of
“eventually
returning
to
scheme
theory”
—
e.g.,
of
showing
that
some
isomor-
phism
of
profinite
groups
arises
from
an
isomorphism
of
schemes
—
is
no
longer
an
issue
of
primary
interest.
One
aspect
of
this
new
point
of
view
is
that
the
main
results
obtained
are
no
longer
necessarily
of
the
form
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
I
3
(∗)
“some
scheme
is
anabelian”
—
i.e.,
some
sort
of
“fundamental
group
functor
X
→
Π
X
”
from
varieties
to
profinite
groups
is
fully
faithful
—
but
rather
of
the
form
(†)
“some
a
priori
scheme-theoretic
property/construction/operation
may
be
formulated
as
a
group-theoretic
algorithm”,
i.e.,
an
algorithm
that
depends
only
on
the
topological
group
structure
of
the
arithmetic
fun-
damental
groups
involved
—
cf.,
e.g.,
(2),
(4)
below.
A
sort
of
intermediate
variant
between
(∗)
and
(†)
is
constituted
by
results
of
the
form
(∗
)
“homomorphisms
between
arithmetic
fundamental
groups
that
satisfy
some
sort
of
relatively
mild
condition
arise
from
scheme
theory”
—
cf.,
e.g.,
(3)
below.
Here,
we
note
that
typically
results
in
absolute
or
semi-absolute
anabelian
geometry
are
much
more
difficult
to
obtain
than
corresponding
results
in
relative
anabelian
geometry
[cf.,
e.g.,
the
discussion
of
(i)
below].
This
is
one
reason
why
one
is
frequently
obliged
to
content
oneself
with
results
of
the
form
(†)
or
(∗
),
as
opposed
to
(∗).
On
the
other
hand,
another
aspect
of
this
new
point
of
view
is
that,
by
abol-
ishing
the
restriction
that
one
must
have
as
one’s
ultimate
goal
the
complete
recon-
struction
of
the
original
schemes
involved,
one
gains
a
greater
degree
of
freedom
in
the
geometries
that
one
considers.
This
greater
degree
of
freedom
often
results
in
the
discovery
of
new
results
that
might
have
eluded
one’s
attention
if
one
restricts
oneself
to
obtaining
results
of
the
form
(∗).
Indeed,
this
phenomenon
may
already
be
seen
in
previous
work
of
the
author:
(i)
In
[Mzk6],
Proposition
1.2.1
[and
its
proof],
various
group-theoretic
al-
gorithms
are
given
for
constructing
various
objects
associated
to
the
ab-
solute
Galois
group
of
a
mixed-characteristic
local
field.
In
this
case,
we
recall
that
it
is
well-known
[cf.,
e.g.,
[NSW],
the
Closing
Remark
preceding
Theorem
12.2.7]
that
in
general,
there
exist
isomorphisms
between
such
absolute
Galois
groups
that
do
not
arise
from
scheme
theory.
(ii)
In
the
theory
of
pro-l
cuspidalizations
given
in
[Mzk13],
§3,
“cuspi-
dalized
geometrically
pro-l
fundamental
groups”
are
“group-theoretically
constructed”
from
geometrically
pro-l
fundamental
groups
of
proper
hy-
perbolic
curves
without
ever
addressing
the
issue
of
whether
or
not
the
original
curve
[i.e.,
scheme]
may
be
reconstructed
from
the
given
geomet-
rically
pro-l
fundamental
group
[of
a
proper
hyperbolic
curve].
(iii)
In
some
sense,
the
abstract,
algorithmic
point
of
view
discussed
above
is
taken
even
further
in
[Mzk12],
where
one
works
with
certain
types
of
purely
combinatorial
objects
—
i.e.,
“semi-graphs
of
anabelioids”
—
whose
definition
“just
happens
to
be”
motivated
by
stable
curves
in
algebraic
ge-
ometry.
On
the
other
hand,
the
results
obtained
in
[Mzk12]
are
results
4
SHINICHI
MOCHIZUKI
concerning
the
abstract
combinatorial
geometry
of
these
abstract
combina-
torial
objects
—
i.e.,
one
is
never
concerned
with
the
issue
of
“eventually
returning”
to,
for
instance,
scheme-theoretic
morphisms.
The
main
results
of
the
present
paper
are,
to
a
substantial
extent,
“generalities”
that
will
be
of
use
to
us
in
the
further
development
of
the
theory
in
the
latter
two
papers
of
the
present
three-part
series.
These
main
results
center
around
the
theme
of
understanding
the
gap
between
relative,
semi-absolute,
and
absolute
anabelian
geometry
and
may
be
summarized
as
follows:
(1)
In
§1,
we
study
various
notions
associated
to
abstract
profinite
groups
such
as
RTF-quotients
[i.e.,
quotients
obtained
by
successive
formation
of
torsion-free
abelianizations
—
cf.
Definition
1.1,
(i)],
slimness
[i.e.,
the
property
that
all
open
subgroups
are
center-free],
and
elasticity
[i.e.,
the
property
that
every
nontrivial
topologically
finitely
generated
closed
normal
subgroup
of
an
open
subgroup
is
itself
open
—
cf.
Definition
1.1,
(ii)]
in
the
context
of
the
absolute
Galois
groups
that
typically
appear
in
anabelian
geometry
[cf.
Proposition
1.5,
Theorem
1.7].
(2)
In
§2,
we
begin
by
formulating
the
terminology
that
we
shall
use
in
our
discussion
of
the
anabelian
geometry
of
quite
general
varieties
of
arbitrary
dimension
[cf.
Definition
2.1].
We
then
apply
the
theory
of
slimness
and
elasticity
developed
in
§1
to
study
various
variants
of
the
notion
of
“semi-
absoluteness”
[cf.
Proposition
2.5].
Moreover,
in
the
case
of
quite
general
varieties
of
arbitrary
dimension
over
number
fields,
mixed-characteristic
local
fields,
or
finite
fields,
we
combine
the
various
group-theoretic
consid-
erations
of
(1)
with
the
classical
theory
of
Albanese
varieties
[reviewed
in
the
Appendix]
to
give
various
“group-theoretic
algorithms”
for
constructing
the
quotient
of
an
arithmetic
fundamental
group
determined
by
the
absolute
Galois
group
of
the
base
field
[cf.
Theorem
2.6,
Corollary
2.8].
Finally,
in
the
case
of
hyperbolic
orbicurves,
we
apply
the
theory
of
max-
imal
pro-RTF-quotients
developed
in
§1
to
give
quite
explicit
“group-
theoretic
algorithms”
for
constructing
these
quotients
[cf.
Theorem
2.11].
Such
maximal
pro-RTF-quotients
may
be
thought
of
as
a
sort
of
analogue,
in
the
case
of
mixed-characteristic
local
fields,
of
the
reconstruction,
in
the
case
of
finite
fields,
of
the
quotient
of
an
arithmetic
fundamental
group
determined
by
the
absolute
Galois
group
of
the
base
field
via
the
opera-
tion
of
“passing
to
the
maximal
torsion-free
abelian
quotient”
[cf.
Remark
2.11.1].
(3)
In
§3,
we
develop
a
generalization
of
the
main
result
of
[Mzk1]
concerning
the
geometricity
of
arbitrary
isomorphisms
of
absolute
Ga-
lois
groups
of
mixed-characteristic
local
fields
that
pre-
serve
the
ramification
filtration
[cf.
Theorem
3.5].
This
generalization
allows
one
to
replace
the
condition
of
“preserving
the
ramification
filtration”
by
various
more
general
conditions,
certain
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
I
5
of
which
were
motivated
by
a
result
orally
communicated
to
the
author
by
A.
Tamagawa
[cf.
Remark
3.8.1].
Moreover,
unlike
the
main
result
of
[Mzk1],
this
generalization
may
be
applied,
in
certain
cases,
to
arbitrary
open
homomorphisms
—
i.e.,
not
just
isomor-
phisms!
—
between
absolute
Galois
groups
of
mixed-characteristic
local
fields,
hence
implies
certain
semi-absolute
Hom-versions
[cf.
Corollary
3.8,
3.9]
of
the
relative
Hom-versions
of
the
Grothendieck
Conjecture
given
in
[Mzk3],
Theorems
A,
B.
Also,
we
observe,
in
Example
2.13,
that
the
corresponding
absolute
Hom-version
of
these
results
is
false
in
general.
Indeed,
it
was
precisely
the
discovery
of
this
counterexample
to
the
“absolute
Hom-
version”
that
led
the
author
to
the
detailed
investigation
of
the
“gap
be-
tween
absolute
and
semi-absolute”
that
forms
the
content
of
§2.
(4)
In
§4,
we
study
various
“fundamental
operations”
for
passing
from
one
algebraic
stack
to
another.
In
the
case
of
arbitrary
dimension,
these
op-
erations
are
the
operations
of
“passing
to
a
finite
étale
covering”
and
“passing
to
a
finite
étale
quotient”;
in
the
case
of
hyperbolic
orbicurves,
we
also
consider
the
operations
of
“forgetting
a
cusp”
and
“coarsifying
a
non-scheme-like
point”.
Our
main
result
asserts
that
if
one
assumes
certain
relative
anabelian
results
concerning
the
varieties
under
consideration,
then
there
exist
group-theoretic
algorithms
for
describing
the
corresponding
semi-absolute
an-
abelian
operations
on
arithmetic
fundamental
groups
[cf.
The-
orem
4.7].
This
theory,
which
generalizes
the
theory
of
[Mzk9],
§2,
and
[Mzk13],
§2,
may
be
applied
not
only
to
hyperbolic
orbicurves
over
sub-p-adic
fields
[cf.
Example
4.8],
but
also
to
“iso-poly-hyperbolic
orbisurfaces”
over
sub-p-
adic
fields
[cf.
Example
4.9].
In
[Mzk15],
this
theory
will
be
applied,
in
an
essential
way,
in
our
development
of
the
theory
of
Belyi
and
elliptic
cusp-
idalizations.
We
also
give
a
tempered
version
of
this
theory
[cf.
Theorem
4.12].
Finally,
in
an
Appendix,
we
review,
for
lack
of
an
appropriate
reference,
various
well-
known
facts
concerning
the
theory
of
Albanese
varieties
that
will
play
an
important
role
in
the
portion
of
the
theory
of
§2
concerning
varieties
of
arbitrary
dimension.
Much
of
this
theory
of
Albanese
varieties
is
contained
in
such
classical
references
as
[NS],
[Serre1],
[Chev],
which
are
written
from
a
somewhat
classical
point
of
view.
Thus,
in
the
Appendix,
we
give
a
modern
scheme-theoretic
treatment
of
this
classical
theory,
but
without
resorting
to
the
introduction
of
motives
and
derived
categories,
as
in
[BS],
[SS].
In
fact,
strictly
speaking,
in
the
proofs
that
appear
in
the
body
of
the
text
[i.e.,
§2],
we
shall
only
make
essential
use
of
the
portion
of
the
Appendix
concerning
abelian
Albanese
varieties
[i.e.,
as
opposed
to
semi-
abelian
Albanese
varieties].
Nevertheless,
we
decided
to
give
a
full
treatment
of
the
theory
of
Albanese
varieties
as
given
in
the
Appendix,
since
it
seemed
to
the
author
that
the
theory
is
not
much
more
difficult
and,
moreover,
assumes
a
much
6
SHINICHI
MOCHIZUKI
more
natural
form
when
formulated
for
“open”
[i.e.,
not
necessarily
proper]
varieties
[which,
roughly
speaking,
correspond
to
the
case
of
semi-abelian
Albanese
varieties]
than
when
formulated
only
for
proper
varieties
[which,
roughly
speaking,
correspond
to
the
case
of
abelian
Albanese
varieties].
Acknowledgements:
I
would
like
to
thank
Akio
Tamagawa
for
many
helpful
discussions
concerning
the
material
presented
in
this
paper.
Also,
I
would
like
to
thank
Brian
Conrad
for
informing
me
of
the
references
in
the
Appendix
to
[FGA],
and
Noboru
Nakayama
for
advice
concerning
non-smooth
normal
algebraic
varieties.
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
I
7
Section
0:
Notations
and
Conventions
Numbers:
The
notation
Q
will
be
used
to
denote
the
field
of
rational
numbers.
The
notation
Z
⊆
Q
will
be
used
to
denote
the
set,
group,
or
ring
of
rational
integers.
The
notation
N
⊆
Z
will
be
used
to
denote
the
set
or
monoid
of
nonnegative
rational
Write
integers.
The
profinite
completion
of
the
group
Z
will
be
denoted
Z.
Primes
for
the
set
of
prime
numbers.
If
p
∈
Primes,
then
the
notation
Q
p
(respectively,
Z
p
)
will
be
used
to
denote
the
p-adic
completion
of
Q
(respectively,
Z).
Also,
we
shall
write
Z
p
(×)
⊆
Z
×
p
2
×
for
the
subgroup
1
+
pZ
p
⊆
Z
×
p
if
p
>
2,
1
+
p
Z
p
⊆
Z
p
if
p
=
2.
Thus,
we
have
isomorphisms
of
topological
groups
∼
(×)
×
Z
(×)
×
(Z
×
p
p
/Z
p
)
→
Z
p
;
∼
Z
p
(×)
→
Z
p
—
where
the
second
isomorphism
is
the
isomorphism
determined
by
dividing
the
(×)
∼
→
F
×
p-adic
logarithm
by
p
if
p
>
2,
or
by
p
2
if
p
=
2;
Z
×
p
/Z
p
p
if
p
>
2,
(×)
∼
Z
×
p
/Z
p
→
Z/pZ
if
p
=
2.
A
finite
field
extension
of
Q
will
be
referred
to
as
a
number
field,
or
NF,
for
short.
A
finite
field
extension
of
Q
p
for
some
p
∈
Primes
will
be
referred
to
as
a
mixed-characteristic
nonarchimedean
local
field,
or
MLF,
for
short.
A
field
of
finite
cardinality
will
be
referred
to
as
a
finite
field,
or
FF,
for
short.
We
shall
regard
the
set
of
symbols
{NF,
MLF,
FF}
as
being
equipped
with
a
linear
ordering
NF
>
MLF
>
FF
and
refer
to
an
element
of
this
set
of
symbols
as
a
field
type.
Topological
Groups:
Let
G
be
a
Hausdorff
topological
group,
and
H
⊆
G
a
closed
subgroup.
Let
us
write
def
Z
G
(H)
=
{g
∈
G
|
g
·
h
=
h
·
g,
∀
h
∈
H}
for
the
centralizer
of
H
in
G;
def
N
G
(H)
=
{g
∈
G
|
g
·
H
·
g
−1
=
H}
for
the
normalizer
of
H
in
G;
and
def
C
G
(H)
=
{g
∈
G
|
(g
·
H
·
g
−1
)
H
has
finite
index
in
H,
g
·
H
·
g
−1
}
for
the
commensurator
of
H
in
G.
Note
that:
(i)
Z
G
(H),
N
G
(H)
and
C
G
(H)
are
subgroups
of
G;
(ii)
we
have
inclusions
H,
Z
G
(H)
⊆
N
G
(H)
⊆
C
G
(H);
(iii)
H
is
8
SHINICHI
MOCHIZUKI
normal
in
N
G
(H).
If
H
=
N
G
(H)
(respectively,
H
=
C
G
(H)),
then
we
shall
say
that
H
is
normally
terminal
(respectively,
commensurably
terminal)
in
G.
Note
that
Z
G
(H),
N
G
(H)
are
always
closed
in
G,
while
C
G
(H)
is
not
necessarily
closed
def
in
G.
Also,
we
shall
write
Z(G)
=
Z
G
(G)
for
the
center
of
G.
Let
G
be
a
topological
group.
Then
[cf.
[Mzk14],
§0]
we
shall
refer
to
a
normal
open
subgroup
H
⊆
G
such
that
the
quotient
group
G/H
is
a
free
discrete
group
as
co-free.
We
shall
refer
to
a
co-free
subgroup
H
⊆
G
as
minimal
if
every
co-free
subgroup
of
G
contains
H.
Thus,
any
minimal
co-free
subgroup
of
G
is
necessarily
unique
and
characteristic.
We
shall
refer
to
a
continuous
homomorphism
between
topological
groups
as
dense
(respectively,
of
DOF-type
[cf.
[Mzk10],
Definition
6.2,
(iii)];
of
OF-type)
if
its
image
is
dense
(respectively,
dense
in
some
open
subgroup
of
finite
index;
an
open
subgroup
of
finite
index).
Let
Π
be
a
topological
group;
Δ
a
normal
closed
subgroup
such
that
every
characteristic
open
subgroup
of
finite
index
H
⊆
Δ
admits
for
the
profinite
completion
of
Π.
a
minimal
co-free
subgroup
H
co-fr
⊆
H.
Write
Π
Let
Q
Π
be
a
quotient
of
profinite
groups.
Then
we
shall
refer
to
as
the
(Q,
Δ)-co-free
]
Q
completion
of
Π,
or
co-free
completion
of
Π
with
respect
to
[the
quotient
Π
and
[the
subgroup]
Δ
⊆
Π
—
where
we
shall
often
omit
mention
of
Δ
when
it
is
fixed
throughout
the
discussion
—
the
inverse
limit
co-fr
Π
Q/co-fr
=
lim
)
←−
Im
Q
(Π/H
def
H
—
where
H
⊆
Δ
ranges
over
the
characteristic
open
subgroups
of
Δ
of
finite
index;
co-fr
⊆
Π
is
the
closure
of
the
image
of
H
co-fr
in
Π;
H
co-fr
⊆
Q
is
the
image
of
H
Q
co-fr
in
Q;
“Im
Q
(−)”
denotes
the
image
in
Q/
H
co-fr
of
the
group
in
parentheses.
H
Q
Thus,
we
have
a
natural
dense
homomorphism
Π
→
Π
Q/co-fr
.
We
shall
say
that
a
profinite
group
G
is
slim
if
for
every
open
subgroup
H
⊆
G,
the
centralizer
Z
G
(H)
is
trivial.
Note
that
every
finite
normal
closed
subgroup
N
⊆
G
of
a
slim
profinite
group
G
is
trivial.
[Indeed,
this
follows
by
observing
that
for
any
normal
open
subgroup
H
⊆
G
such
that
N
H
=
{1},
consideration
of
the
inclusion
N
→
G/H
reveals
that
the
conjugation
action
of
H
on
N
is
trivial,
i.e.,
that
N
⊆
Z
G
(H)
=
{1}.]
We
shall
say
that
a
profinite
group
G
is
decomposable
if
there
exists
an
iso-
∼
morphism
of
profinite
groups
H
1
×
H
2
→
G,
where
H
1
,
H
2
are
nontrivial
profinite
groups.
If
a
profinite
group
G
is
not
decomposable,
then
we
shall
say
that
it
is
indecomposable.
We
shall
write
G
ab
for
the
abelianization
of
a
profinite
group
G,
i.e.,
the
quo-
tient
of
G
by
the
closure
of
the
commutator
subgroup
of
G,
and
G
ab-t
for
the
torsion-free
abelianization
of
G,
i.e.,
the
quotient
of
G
ab
by
the
closure
of
the
torsion
subgroup
of
G
ab
.
Note
that
the
formation
of
G
ab
,
G
ab-t
is
functorial
with
respect
to
arbitrary
continuous
homomorphisms
of
profinite
groups.
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
I
9
We
shall
denote
the
group
of
automorphisms
of
a
profinite
group
G
by
Aut(G).
Conjugation
by
elements
of
G
determines
a
homomorphism
G
→
Aut(G)
whose
image
consists
of
the
inner
automorphisms
of
G.
We
shall
denote
by
Out(G)
the
quotient
of
Aut(G)
by
the
[normal]
subgroup
consisting
of
the
inner
auto-
morphisms.
In
particular,
if
G
is
center-free,
then
we
have
an
exact
sequence
1
→
G
→
Aut(G)
→
Out(G)
→
1.
If,
moreover,
G
is
topologically
finitely
generated,
then
it
follows
immediately
that
the
topology
of
G
admits
a
basis
of
characteristic
open
subgroups,
which
thus
determine
a
topology
on
Aut(G),
Out(G)
with
respect
to
which
the
exact
sequence
1
→
G
→
Aut(G)
→
Out(G)
→
1
may
be
regarded
as
an
exact
sequence
of
profinite
groups.
Algebraic
Stacks
and
Log
Schemes:
We
refer
to
[FC],
Chapter
I,
§4.10,
for
a
discussion
of
the
coarse
space
associated
to
an
algebraic
stack.
We
shall
say
that
an
algebraic
stack
is
scheme-like
if
it
is,
in
fact,
a
scheme.
We
shall
say
that
an
algebraic
stack
is
generically
scheme-like
if
it
admits
an
open
dense
substack
which
is
a
scheme.
We
refer
to
[Kato]
and
the
references
given
in
[Kato]
for
basic
facts
and
def-
initions
concerning
log
schemes.
Here,
we
recall
that
the
interior
of
a
log
scheme
refers
to
the
largest
open
subscheme
over
which
the
log
structure
is
trivial.
Curves:
We
shall
use
the
following
terms,
as
they
are
defined
in
[Mzk13],
§0:
hyperbolic
curve,
family
of
hyperbolic
curves,
cusp,
tripod.
Also,
we
refer
to
[Mzk6],
the
proof
of
Lemma
2.1;
[Mzk6],
the
discussion
following
Lemma
2.1,
for
an
explanation
of
the
terms
“stable
reduction”
and
“stable
model”
applied
to
a
hyperbolic
curve
over
an
MLF.
If
X
is
a
generically
scheme-like
algebraic
stack
over
a
field
k
that
admits
a
finite
étale
Galois
covering
Y
→
X,
where
Y
is
a
hyperbolic
curve
over
a
finite
extension
of
k,
then
we
shall
refer
to
X
as
a
hyperbolic
orbicurve
over
k.
[Thus,
when
k
is
of
characteristic
zero,
this
definition
coincides
with
the
definition
of
a
“hyperbolic
orbicurve”
in
[Mzk13],
§0,
and
differs
from,
but
is
equivalent
to,
the
definition
of
a
“hyperbolic
orbicurve”
given
in
[Mzk7],
Definition
2.2,
(ii).
We
refer
to
[Mzk13],
§0,
for
more
on
this
equivalence.]
Note
that
the
notion
of
a
“cusp
of
a
hyperbolic
curve”
given
in
[Mzk13],
§0,
generalizes
immediately
to
the
notion
of
“cusp
of
a
hyperbolic
orbicurve”.
If
X
→
Y
is
a
dominant
morphism
of
hyperbolic
orbicurves,
then
we
shall
refer
to
X
→
Y
as
a
partial
coarsification
morphism
if
the
morphism
induced
by
X
→
Y
on
associated
coarse
spaces
is
an
isomorphism.
Let
X
be
a
hyperbolic
orbicurve
over
an
algebraically
closed
field;
denote
its
étale
fundamental
group
by
Δ
X
.
We
shall
refer
to
the
order
of
the
[manifestly
finite!]
decomposition
group
in
Δ
X
of
a
closed
point
x
of
X
as
the
order
of
x.
We
shall
refer
to
the
[manifestly
finite!]
least
common
multiple
of
the
orders
of
the
closed
points
of
X
as
the
order
of
X.
Thus,
it
follows
immediately
from
the
definitions
that
X
is
a
hyperbolic
curve
if
and
only
if
the
order
of
X
is
equal
to
1.
10
SHINICHI
MOCHIZUKI
Section
1:
Some
Profinite
Group
Theory
We
begin
by
discussing
certain
aspects
of
abstract
profinite
groups,
as
they
relate
to
the
Galois
groups
of
finite
fields,
mixed-characteristic
nonarchimedean
local
fields,
and
number
fields.
In
the
following,
let
G
be
a
profinite
group.
Definition
1.1.
(i)
In
the
following,
“RTF”
is
to
be
understood
as
an
abbreviation
for
“recur-
sively
torsion-free”.
If
H
⊆
G
is
a
normal
open
subgroup
that
arises
as
the
kernel
of
a
continuous
surjection
G
Q,
where
Q
is
a
finite
abelian
group,
that
factors
through
the
torsion-free
abelianization
G
G
ab-t
of
G
[cf.
§0],
then
we
shall
refer
to
(G,
H)
as
an
RTF-pair.
If
for
some
integer
n
≥
1,
a
sequence
of
open
subgroups
G
n
⊆
G
n−1
⊆
.
.
.
G
1
⊆
G
0
=
G
of
G
satisfies
the
condition
that,
for
each
nonnegative
integer
j
≤
n
−
1,
(G
j
,
G
j+1
)
is
an
RTF-pair,
then
we
shall
refer
to
this
sequence
of
open
subgroups
as
an
RTF-
chain
[from
G
n
to
G].
If
H
⊆
G
is
an
open
subgroup
such
that
there
exists
an
RTF-chain
from
H
to
G,
then
we
shall
refer
to
H
⊆
G
as
an
RTF-subgroup
[of
G].
If
the
kernel
of
a
continuous
surjection
φ
:
G
Q,
where
Q
is
a
finite
group,
is
an
RTF-subgroup
of
G,
then
we
shall
say
that
φ
:
G
Q
is
an
RTF-quotient
of
G.
If
φ
:
G
Q
is
a
continuous
surjection
of
profinite
groups
such
that
the
topology
of
Q
admits
a
basis
of
normal
open
subgroups
{N
α
}
α∈A
satisfying
the
property
that
each
composite
G
Q
Q/N
α
[for
α
∈
A]
is
an
RTF-quotient,
then
we
shall
say
that
φ
:
G
Q
is
a
pro-RTF-quotient.
If
G
is
a
profinite
group
such
that
the
identity
map
of
G
forms
a
pro-RTF-quotient,
then
we
shall
say
that
G
is
a
pro-RTF-group.
[Thus,
every
pro-RTF-group
is
pro-solvable.]
(ii)
We
shall
say
that
G
is
elastic
if
it
holds
that
every
topologically
finitely
generated
closed
normal
subgroup
N
⊆
H
of
an
open
subgroup
H
⊆
G
of
G
is
either
trivial
or
of
finite
index
in
G.
If
G
is
elastic,
but
not
topologically
finitely
generated,
then
we
shall
say
that
G
is
very
elastic.
(iii)
Let
Σ
⊆
Primes
[cf.
§0]
be
a
set
of
prime
numbers.
If
G
admits
an
open
subgroup
which
is
pro-Σ,
then
we
shall
say
that
G
is
almost
pro-Σ.
We
shall
refer
to
a
quotient
G
Q
as
almost
pro-Σ-maximal
if
for
some
normal
open
subgroup
N
⊆
G
with
maximal
pro-Σ
quotient
N
P
,
we
have
Ker(G
Q)
=
Ker(N
P
).
[Thus,
any
almost
pro-Σ-maximal
quotient
of
G
is
almost
pro-Σ.]
def
If
Σ
=
Primes
\
{p}
for
some
p
∈
Primes,
then
we
shall
write
“pro-(
=
p)”
for
“pro-Σ”.
Write
(
=
p)
Z
We
shall
say
that
G
is
pro-omissive
(re-
for
the
maximal
pro-(
=
p)
quotient
of
Z.
spectively,
almost
pro-omissive)
if
it
is
pro-(
=
p)
for
some
p
∈
Primes
(respectively,
if
it
admits
a
pro-omissive
open
subgroup).
We
shall
say
that
G
is
augmented
pro-p
(
=
p)
→
1,
if
there
exists
an
exact
sequence
of
profinite
groups
1
→
N
→
G
→
Z
where
N
is
pro-p;
in
this
case,
the
image
of
N
in
G
is
uniquely
determined
[i.e.,
as
the
(
=
p)
[which
is
well-defined
up
to
maximal
pro-p
subgroup
of
G];
the
quotient
G
Z
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
I
11
(
=
p)
]
will
be
referred
to
as
the
augmentation
of
the
augmented
automorphisms
of
Z
pro-p
group
G.
We
shall
say
that
G
is
augmented
pro-prime
if
it
is
augmented
pro-
p
for
some
[not
necessarily
unique!]
p
∈
Primes.
If
Σ
=
{p}
for
some
unspecified
p
∈
Primes,
we
shall
write
“pro-prime”
for
“pro-Σ”.
If
C
is
the
“full
formation”
[cf.,
e.g.,
[FJ],
p.
343]
of
finite
solvable
Σ-groups,
then
we
shall
refer
to
a
pro-C
group
as
a
pro-Σ-solvable
group.
Proposition
1.2.
(Basic
Properties
of
Pro-RTF-quotients)
Let
φ
:
G
1
→
G
2
be
a
continuous
homomorphism
of
profinite
groups.
Then:
(i)
If
H
⊆
G
2
is
an
RTF-subgroup
of
G
2
,
then
φ
−1
(H)
⊆
G
1
is
an
RTF-
subgroup
of
G
1
.
(ii)
If
H,
J
⊆
G
are
RTF-subgroups
of
G,
then
so
is
H
J.
(iii)
If
H
⊆
G
is
an
RTF-subgroup
of
G,
then
there
exists
a
normal
[open]
RTF-subgroup
J
⊆
G
of
G
such
that
J
⊆
H.
(iv)
Every
RTF-quotient
G
Q
of
G
factors
through
the
quotient
def
G
G
RTF
=
←
lim
−
G/N
N
—
where
N
ranges
over
the
normal
[open]
RTF-subgroups
of
G.
We
shall
refer
to
this
quotient
G
G
RTF
as
the
maximal
pro-RTF-quotient.
Finally,
the
profinite
group
G
RTF
is
a
pro-RTF-group.
(v)
There
exists
a
commutative
diagram
G
1
⏐
⏐
G
RTF
1
φ
−→
φ
RTF
−→
G
2
⏐
⏐
G
RTF
2
—
where
the
vertical
arrows
are
the
natural
morphisms,
and
the
continuous
homo-
morphism
φ
RTF
is
uniquely
determined
by
the
condition
that
the
diagram
commute.
Proof.
Assertion
(i)
follows
immediately
from
the
definitions,
together
with
the
functoriality
of
the
torsion-free
abelianization
[cf.
§0].
To
verify
assertion
(ii),
one
observes
that
an
RTF-chain
from
H
J
to
G
may
be
obtained
by
concatenating
an
RTF-chain
from
H
J
to
J
[whose
existence
follows
from
assertion
(i)
applied
to
the
natural
inclusion
homomorphism
J
→
G]
with
an
RTF-chain
from
J
to
G.
Assertion
(iii)
follows
by
applying
assertion
(ii)
to
some
finite
intersection
of
conjugates
of
H.
Assertion
(iv)
follows
immediately
from
assertions
(ii),
(iii),
and
the
definitions
involved.
Assertion
(v)
follows
immediately
from
assertions
(i),
(iv).
12
SHINICHI
MOCHIZUKI
Proposition
1.3.
(Basic
Properties
of
Elasticity)
(i)
Let
H
⊆
G
be
an
open
subgroup
of
the
profinite
group
G.
Then
the
elas-
ticity
of
G
implies
that
of
H.
If
G
is
slim,
then
the
elasticity
of
H
implies
that
of
G.
(ii)
Suppose
that
G
is
nontrivial.
Then
G
is
very
elastic
if
and
only
if
it
holds
that
every
topologically
finitely
generated
closed
normal
subgroup
N
⊆
H
of
an
open
subgroup
H
⊆
G
of
G
is
trivial.
Proof.
Assertion
(i)
follows
immediately
from
the
definitions,
together
with
the
fact
that
a
slim
profinite
group
has
no
normal
closed
finite
subgroups
[cf.
§0].
The
necessity
portion
of
assertion
(ii)
follows
from
the
fact
that
the
existence
of
a
topo-
logically
finitely
generated
open
subgroup
of
G
implies
that
G
itself
is
topologically
finitely
generated;
the
sufficiency
portion
of
assertion
(ii)
follows
immediately
by
def
taking
N
=
G
=
{1}.
Next,
we
consider
Galois
groups.
Definition
1.4.
We
shall
refer
to
a
field
k
as
solvably
closed
if,
for
every
finite
abelian
field
extension
k
of
k,
it
holds
that
k
=
k.
Remark
1.4.1.
Note
that
if
k
is
a
solvably
closed
Galois
extension
of
a
field
k
of
type
MLF
or
FF
[cf.
§0],
then
k
is
an
algebraic
closure
of
k.
Indeed,
this
follows
from
the
well-known
fact
that
the
absolute
Galois
group
of
a
field
of
type
MLF
or
FF
is
pro-solvable
[cf.,
e.g.,
[NSW],
Chapter
VII,
§5].
Proposition
1.5.
(Pro-RTF-quotients
of
MLF
Galois
Groups)
Let
k
be
an
def
algebraic
closure
of
an
MLF
[cf.
§0]
k
of
residue
characteristic
p;
G
k
=
Gal(k/k);
G
k
G
RTF
the
maximal
pro-RTF-quotient
[cf.
Proposition
1.2,
(iv)]
of
G
k
.
k
Then:
(i)
G
RTF
is
slim.
k
→
1,
where
P
is
a
(ii)
There
exists
an
exact
sequence
1
→
P
→
G
RTF
→
Z
k
RTF
pro-p
group
whose
image
in
G
k
is
equal
to
the
image
of
the
inertia
subgroup
RTF
RTF
of
G
k
in
G
k
.
In
particular,
G
k
is
augmented
pro-p.
Proof.
Recall
from
local
class
field
theory
[cf.,
e.g.,
[Serre2]]
that
for
any
open
sub-
group
H
⊆
G
k
,
corresponding
to
a
subfield
k
H
⊆
k,
we
have
a
natural
isomorphism
∼
×
∧
)
→
H
ab
(k
H
[where
the
“∧”
denotes
the
profinite
completion
of
an
abelian
group;
“×”
denotes
the
group
of
units
of
a
ring];
moreover,
H
ab
fits
into
an
exact
sequence
→
1
1
→
O
k
×
H
→
H
ab
→
Z
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
I
13
[where
O
k
H
⊆
k
H
is
the
ring
of
integers]
in
which
the
image
of
O
k
×
H
in
H
ab
coincides
with
the
image
of
the
inertia
subgroup
of
H.
Observe,
moreover,
that
the
quotient
of
the
abelian
profinite
group
O
k
×
H
by
its
torsion
subgroup
is
a
pro-p
group.
Thus,
assertion
(ii)
follows
immediately
from
this
observation,
together
with
the
definition
of
the
maximal
pro-RTF-quotient.
Next,
let
us
observe
that
by
applying
the
natural
∼
isomorphism
(O
k
×
H
)
⊗
Q
p
→
k
H
,
it
follows
that
whenever
H
is
normal
in
G
k
,
the
action
of
G
k
/H
on
H
ab-t
is
faithful.
Thus,
assertion
(i)
follows
immediately.
The
following
result
is
well-known.
Proposition
1.6.
(Maximal
Pro-p
Quotients
of
MLF
Galois
Groups)
Let
def
k
be
an
algebraic
closure
of
an
MLF
k
of
residue
characteristic
p;
G
k
=
Gal(k/k);
(p)
G
k
G
k
the
maximal
pro-p-quotient
of
G
k
.
Then:
(i)
Any
almost
pro-p-maximal
quotient
G
k
Q
of
G
k
is
slim.
(ii)
Suppose
further
that
k
contains
a
primitive
p-th
root
of
unity.
Then
(p)
for
any
finite
module
M
annihilated
by
p
equipped
with
a
continuous
action
by
G
k
[which
thus
determines
a
continuous
action
by
G
k
],
the
natural
homomorphism
(p)
G
k
G
k
induces
an
isomorphism
∼
H
j
(G
k
,
M
)
→
H
j
(G
k
,
M
)
(p)
on
Galois
cohomology
modules
for
all
integers
j
≥
0.
(iii)
If
k
contains
(respectively,
does
not
contain)
a
primitive
p-th
root
of
unity,
then
any
closed
subgroup
of
infinite
index
(respectively,
any
closed
(p)
subgroup
of
arbitrary
index)
H
⊆
G
k
is
a
free
pro-p
group.
Proof.
Assertion
(i)
follows
from
the
argument
applied
to
verify
Proposition
1.5,
(i).
To
verify
assertion
(ii),
it
suffices
to
show
that
the
cohomology
module
j
H
j
(J,
M
)
∼
=
lim
−→
H
(G
k
,
M
)
k
def
[where
J
=
Ker(G
k
G
k
);
k
ranges
over
the
finite
Galois
extensions
of
k
such
that
[k
:
k]
is
a
power
of
p;
G
k
⊆
G
k
is
the
open
subgroup
determined
by
k
]
vanishes
for
j
≥
1.
By
“dévissage”,
we
may
assume
that
M
∼
=
F
p
with
the
trivial
G
k
-action.
Since
the
cohomological
dimension
of
G
k
is
equal
to
2
[cf.
[NSW],
Theorem
7.1.8,
(i)],
it
suffices
to
consider
the
cases
j
=
1,
2.
For
j
=
2,
since
H
2
(G
k
,
F
p
)
∼
=
F
p
[cf.
[NSW],
Theorem
7.1.8,
(ii);
our
hypothesis
that
k
contains
a
primitive
p-th
root
of
unity],
it
suffices,
by
the
well-known
functorial
behavior
of
H
2
(G
k
,
F
p
)
[cf.
[NSW],
Corollary
7.1.4],
to
observe
that
k
always
admits
a
cyclic
Galois
extension
of
degree
p
[arising,
for
instance,
from
an
extension
of
the
residue
field
of
k
].
On
the
other
hand,
for
j
=
1,
the
desired
vanishing
is
a
tautology,
(p)
in
light
of
the
definition
of
the
quotient
G
k
G
k
.
This
completes
the
proof
of
assertion
(ii).
(p)
14
SHINICHI
MOCHIZUKI
Finally,
we
consider
assertion
(iii).
If
k
does
not
contain
a
primitive
p-th
root
(p)
of
unity,
then
G
k
itself
is
a
free
pro-p
group
[cf.
[NSW],
Theorem
7.5.8,
(i)],
so
(p)
any
closed
subgroup
H
⊆
G
k
is
also
free
pro-p
[cf.,
e.g.,
[RZ],
Corollary
7.7.5].
Thus,
let
us
assume
that
k
contains
a
primitive
p-th
root
of
unity,
so
we
may
(p)
apply
the
isomorphism
of
assertion
(ii).
In
particular,
if
J
⊆
G
k
is
an
open
subgroup
such
that
H
⊆
J,
and
k
J
⊆
k
is
the
subfield
determined
by
J,
then
one
verifies
immediately
that
the
quotient
G
k
J
J
may
be
identified
with
the
quotient
∼
(p)
G
k
J
G
k
J
,
so
we
obtain
an
isomorphism
H
2
(J,
F
p
)
→
H
2
(G
k
J
,
F
p
)
[where
F
p
is
equipped
with
the
trivial
Galois
action].
Thus,
to
complete
the
proof
that
H
is
free
pro-p,
it
suffices
[by
a
well-known
cohomological
criterion
for
free
pro-p
groups
—
cf.,
e.g.,
[RZ],
Theorem
7.7.4]
to
show
that
the
cohomology
module
2
H
2
(H,
F
p
)
∼
=
lim
−→
H
(G
k
J
,
F
p
)
k
J
[where
F
p
is
equipped
with
the
trivial
Galois
action;
k
J
ranges
over
the
finite
(p)
extensions
of
k
arising
from
open
subgroups
J
⊆
G
k
such
that
H
⊆
J]
vanishes.
As
in
the
proof
of
assertion
(ii),
this
vanishing
follows
from
the
well-known
functorial
behavior
of
H
2
(G
k
J
,
F
p
),
together
with
the
observation
that,
by
our
assumption
that
(p)
H
is
of
infinite
index
in
G
k
,
k
J
always
admits
an
extension
of
degree
p
arising
(p)
from
an
open
subgroup
of
J
[where
J
⊆
G
k
corresponds
to
k
J
]
containing
H.
Theorem
1.7.
(Slimness
and
Elasticity
of
Arithmetic
Galois
Groups)
def
k/k).
Let
k
be
a
solvably
closed
Galois
extension
of
a
field
k;
write
G
k
=
Gal(
Then:
is
neither
elastic
nor
slim.
(i)
If
k
is
an
FF,
then
G
k
∼
=
Z
(ii)
If
k
is
an
MLF
of
residue
characteristic
p,
then
G
k
,
as
well
as
any
almost
pro-p-maximal
quotient
G
k
Q
of
G
k
,
is
elastic
and
slim.
(iii)
If
k
is
an
NF,
then
G
k
is
very
elastic
and
slim.
Proof.
Assertion
(i)
is
immediate
from
the
definitions;
assertion
(iii)
is
the
content
of
[Mzk11],
Corollary
2.2;
[Mzk11],
Theorem
2.4.
The
slimness
portion
of
assertion
(ii)
for
G
k
is
shown,
for
instance,
in
[Mzk6],
Theorem
1.1.1,
(ii)
[via
the
same
argument
as
the
argument
applied
to
prove
Proposition
1.5,
(i);
Proposition
1.6,
(i)];
the
slimness
portion
of
assertion
(ii)
for
Q
is
precisely
the
content
of
Proposition
1.6,
(i).
Write
p
for
the
residue
characteristic
of
k.
To
show
the
elasticity
portion
of
assertion
(ii)
for
Q,
let
N
⊆
H
be
a
closed
normal
subgroup
of
infinite
index
of
an
open
subgroup
H
⊆
Q
such
that
N
is
topologically
generated
by
r
elements,
where
r
≥
1
is
an
integer.
Then
it
suffices
to
show
that
N
is
trivial.
Since
Q
has
already
been
shown
to
be
slim
[hence
has
no
nontrivial
finite
normal
closed
subgroups
—
cf.
§0],
we
may
always
replace
k
by
a
finite
extension
of
k.
In
particular,
we
may
assume
that
H
=
Q,
and
that
Q
is
maximal
pro-p.
Since
[Q
:
N
]
is
infinite,
it
follows
that
there
exists
an
open
subgroup
J
⊆
Q,
corresponding
to
a
subfield
k
J
⊆
k,
such
that
N
⊆
J,
and
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
I
15
[k
J
:
Q
p
]
≥
r
+
1.
Here,
we
recall
from
our
discussion
of
local
class
field
theory
in
the
proof
of
Proposition
1.5
that
dim
Q
p
(J
ab
⊗
Q
p
)
=
[k
J
:
Q
p
]
+
1
(≥
r
+
2).
In
particular,
we
conclude
that
N
is
necessarily
a
subgroup
of
infinite
index
of
some
topologically
finitely
generated
closed
subgroup
P
⊆
J
such
that
[J
:
P
]
is
infinite.
[For
instance,
one
may
take
P
to
be
the
subgroup
of
J
topologically
generated
by
N
,
together
with
an
element
of
J
that
maps
to
a
non-torsion
element
of
the
quotient
of
J
ab
by
the
image
of
N
ab
.]
Thus,
we
conclude
from
Proposition
1.6,
(iii),
that
P
is
a
free
pro-p
group
which
contains
a
topologically
finitely
generated
closed
normal
subgroup
N
⊆
P
of
infinite
index.
On
the
other
hand,
by
[a
rather
easy
special
case
of]
the
theorem
of
Lubotzky-Melnikov-van
den
Dries
[cf.,
e.g.,
[FJ],
Proposition
24.10.3;
[MT],
Theorem
1.5],
this
implies
that
N
is
trivial.
This
completes
the
proof
of
the
elasticity
portion
of
assertion
(ii)
for
Q.
To
show
the
elasticity
portion
of
assertion
(ii)
for
G
k
,
let
N
⊆
H
be
a
closed
normal
subgroup
of
infinite
index
of
an
open
subgroup
H
⊆
G
k
such
that
N
is
topologically
generated
by
r
elements,
where
r
≥
1
is
an
integer.
Then
it
suffices
to
show
that
N
is
trivial.
As
in
the
proof
of
the
elasticity
of
“Q”,
we
may
assume
that
H
=
G
k
;
also,
since
[G
k
:
N
]
is
infinite,
by
passing
to
a
finite
extension
of
k
corresponding
to
an
open
subgroup
of
G
k
containing
N
,
we
may
assume
that
[k
:
Q
p
]
≥
r.
But
this
implies
that
the
image
of
N
in
G
ab
k
⊗
Z
p
[which
is
of
rank
[k
:
Q
p
]
+
1
≥
r
+
1]
is
of
infinite
index,
hence
that
the
image
of
N
in
any
almost
pro-p-maximal
quotient
G
k
Q
is
of
infinite
index.
Thus,
by
the
elasticity
of
“Q”,
we
conclude
that
such
images
are
trivial.
Since,
moreover,
the
natural
surjection
G
k
lim
←−
Q
Q
[where
Q
ranges
over
the
almost
pro-p-maximal
quotients
of
G
k
]
is
[by
the
definition
of
the
term
“almost
pro-p-maximal
quotient”]
an
isomorphism,
this
is
enough
to
conclude
that
N
is
trivial,
as
desired.
16
SHINICHI
MOCHIZUKI
Section
2:
Semi-absolute
Anabelian
Geometry
In
the
present
§2,
we
consider
the
problem
of
characterizing
“group-theoretically”
the
quotient
morphism
to
the
Galois
group
of
the
base
field
of
the
arithmetic
fun-
damental
group
of
a
variety.
In
particular,
the
theory
of
the
present
§2
refines
the
theory
of
[Mzk6],
Lemma
1.1.4,
in
two
respects:
We
extend
this
theory
to
the
case
of
quite
general
varieties
of
arbitrary
dimension
[cf.
Corollary
2.8],
and,
in
the
case
of
hyperbolic
orbicurves,
we
give
a
“group-theoretic
version”
of
the
numerical
criterion
of
[Mzk6],
Lemma
1.1.4,
via
the
theory
of
maximal
pro-RTF-quotients
developed
in
§1
[cf.
Corollary
2.12].
The
theory
of
the
present
§2
depends
on
the
general
theory
of
Albanese
varieties,
which
we
review
in
the
Appendix,
for
the
convenience
of
the
reader.
Suppose
that:
(1)
k
is
a
perfect
field,
k
an
algebraic
closure
of
k,
k
⊆
k
a
solvably
closed
def
k/k).
Galois
extension
of
k,
and
G
k
=
Gal(
(2)
X
→
Spec(k)
is
a
geometrically
connected,
smooth,
separated
algebraic
stack
of
finite
type
over
k.
(3)
Y
→
X
is
a
connected
finite
étale
Galois
covering
which
is
a
[necessarily
separated,
smooth,
and
of
finite
type
over
k]
k-scheme
such
that
Gal(Y
/X)
acts
freely
on
some
nonempty
open
subscheme
of
Y
[so
X
is
generically
scheme-like
—
cf.
§0].
(4)
Y
→
Y
is
an
open
immersion
into
a
connected
proper
k-scheme
Y
such
log
that
Y
is
the
underlying
scheme
of
a
log
scheme
Y
that
is
log
smooth
over
k
[where
we
regard
Spec(k)
as
equipped
with
the
trivial
log
structure],
and
the
image
of
Y
in
Y
coincides
with
the
interior
[cf.
§0]
of
the
log
scheme
log
Y
.
Thus,
it
follows
from
the
log
purity
theorem
[which
is
exposed,
for
instance,
in
[Mzk4]
as
“Theorem
B”]
that
the
condition
that
a
finite
étale
covering
Z
→
Y
be
tamely
ramified
over
the
height
one
primes
of
Y
is
equivalent
to
the
condition
log
log
that
the
normalization
Z
of
Y
in
Z
determine
a
log
étale
morphism
Z
→
Y
[whose
underlying
morphism
of
schemes
is
Z
→
Y
];
in
particular,
one
concludes
immediately
that
the
condition
that
Z
→
Y
be
tamely
ramified
over
the
height
one
primes
of
Y
is
independent
of
the
choice
of
“log
smooth
log
compactification”
log
Y
for
Y
.
Thus,
one
verifies
immediately
[by
considering
the
various
Gal(Y
/X)-
log
conjugates
of
the
“log
compactification”
Y
]
that
the
finite
étale
coverings
of
X
whose
pull-backs
to
Y
are
tamely
ramified
over
[the
height
one
primes
of]
Y
form
a
Galois
category,
whose
associated
profinite
group
[relative
to
an
appropriate
choice
of
basepoint
for
X]
we
denote
by
π
1
tame
(X,
Y
),
or
simply
π
1
tame
(X)
when
Y
→
X
is
fixed.
In
particular,
if
we
use
the
subscript
“k”
to
denote
base-
change
from
k
to
k,
then
by
choosing
a
connected
component
of
Y
k
,
we
obtain
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
I
17
a
subgroup
π
1
tame
(X
k
)
⊆
π
1
tame
(X)
which
fits
into
a
natural
exact
sequence
1
→
π
1
tame
(X
k
)
→
π
1
tame
(X)
→
Gal(k/k)
→
1.
Next,
let
Σ
⊆
Primes
be
a
set
of
prime
numbers;
π
1
tame
(X
k
)
Δ
X
an
al-
most
pro-Σ-maximal
quotient
of
π
1
tame
(X
k
)
whose
kernel
is
normal
in
π
1
tame
(X),
hence
determines
a
quotient
π
1
tame
(X)
Π
X
;
we
also
assume
that
the
quotient
π
1
tame
(X)
Gal(Y
/X)
admits
a
factorization
π
1
tame
(X)
Π
X
Gal(Y
/X),
and
that
the
kernel
of
the
resulting
homomorphism
Δ
X
→
Gal(Y
/X)
is
pro-Σ.
Thus,
Ker(Δ
X
→
Gal(Y
/X))
may
be
identified
with
the
maximal
pro-Σ
quotient
of
Ker(π
1
tame
(X
k
)
→
Gal(Y
/X));
we
obtain
a
natural
exact
sequence
1
→
Δ
X
→
Π
X
→
Gal(k/k)
→
1
—
which
may
be
thought
of
as
an
extension
of
the
profinite
group
Gal(k/k).
Definition
2.1.
(i)
We
shall
refer
to
any
profinite
group
Δ
which
is
isomorphic
to
the
profinite
group
Δ
X
constructed
in
the
above
discussion
for
some
choice
of
data
(k,
X,
Y
→
Y
,
Σ)
as
a
profinite
group
of
[almost
pro-Σ]
GFG-type
[where
“GFG”
is
to
be
under-
stood
as
an
abbreviation
for
“geometric
fundamental
group”].
In
this
situation,
we
shall
refer
to
any
surjection
π
1
tame
(X
k
)
Δ
obtained
by
composing
the
surjection
∼
π
1
tame
(X
k
)
Δ
X
with
an
isomorphism
Δ
X
→
Δ
as
a
scheme-theoretic
envelope
for
Δ;
we
shall
refer
to
(k,
X,
Y
→
Y
,
Σ)
as
a
collection
of
construction
data
for
Δ.
[Thus,
given
a
profinite
group
of
GFG-type,
there
are,
in
general,
many
possible
choices
of
construction
data
for
the
profinite
group.]
(ii)
We
shall
refer
to
any
extension
1
→
Δ
→
Π
→
G
→
1
of
profinite
groups
which
is
isomorphic
to
the
extension
1
→
Δ
X
→
Π
X
→
Gal(k/k)
→
1
constructed
in
the
above
discussion
for
some
choice
of
data
(k,
X,
Y
→
Y
,
Σ)
as
an
extension
of
[geometrically
almost
pro-Σ]
AFG-type
[where
“AFG”
is
to
be
understood
as
an
abbreviation
for
“arithmetic
fundamental
group”].
In
this
situation,
we
shall
refer
to
any
surjection
π
1
tame
(X)
Π
(respectively,
any
surjection
π
1
tame
(X
k
)
Δ;
any
∼
isomorphism
Gal(k/k)
→
G)
obtained
by
composing
the
surjection
π
1
tame
(X)
Π
X
(respectively,
the
surjection
π
1
tame
(X
k
)
Δ
X
;
the
identity
Gal(k/k)
=
Gal(k/k))
∼
∼
∼
with
an
isomorphism
Π
X
→
Π
(respectively,
Δ
X
→
Δ;
Gal(k/k)
→
G)
arising
from
an
isomorphism
of
the
extensions
1
→
Δ
→
Π
→
G
→
1,
1
→
Δ
X
→
Π
X
→
Gal(k/k)
→
1
as
a
scheme-theoretic
envelope
for
Π
(respectively,
Δ;
G);
we
shall
refer
to
(k,
X,
Y
→
Y
,
Σ)
as
a
collection
of
construction
data
for
this
extension.
[Thus,
given
an
extension
of
AFG-type,
there
are,
in
general,
many
possible
choices
of
construction
data
for
the
extension.]
(iii)
Let
1
→
Δ
∗
→
Π
∗
→
G
∗
→
1
be
an
extension
of
AFG-type;
N
⊆
G
∗
the
inverse
image
of
the
kernel
of
the
quotient
Gal(k/k)
G
k
relative
to
some
∼
scheme-theoretic
envelope
Gal(k/k)
→
G
∗
.
Suppose
further
that
Δ
∗
is
slim,
and
that
the
outer
action
of
N
on
Δ
∗
[arising
from
the
extension
structure]
is
trivial.
Thus,
every
element
of
N
⊆
G
∗
lifts
to
a
unique
element
of
Π
∗
that
commutes
with
Δ
∗
.
In
particular,
N
lifts
to
a
closed
normal
subgroup
N
Π
⊆
Π
∗
.
We
shall
refer
to
any
extension
1
→
Δ
→
Π
→
G
→
1
of
profinite
groups
which
is
isomorphic
18
SHINICHI
MOCHIZUKI
to
an
extension
of
the
form
1
→
Δ
∗
→
Π
∗
/N
Π
→
G
∗
/N
→
1
just
constructed
as
an
extension
of
[geometrically
almost
pro-Σ]
GSAFG-type
[where
“GSAFG”
is
to
be
understood
as
an
abbreviation
for
“geometrically
slim
arithmetic
fundamental
group”].
In
this
situation,
we
shall
refer
to
any
surjection
π
1
tame
(X)
Π
(respec-
tively,
π
1
tame
(X
k
)
Δ;
Gal(k/k)
G)
obtained
by
composing
a
scheme-theoretic
∼
envelope
π
1
tame
(X)
Π
∗
(respectively,
π
1
tame
(X
k
)
Δ
∗
;
Gal(k/k)
→
G
∗
)
with
the
surjection
Π
∗
Π
(respectively,
Δ
∗
Δ;
G
∗
G)
in
the
above
discus-
sion
as
a
scheme-theoretic
envelope
for
Π
(respectively,
Δ;
G);
we
shall
refer
to
(k,
k,
X,
Y
→
Y
,
Σ)
as
a
collection
of
construction
data
for
this
extension.
[Thus,
given
an
extension
of
GSAFG-type,
there
are,
in
general,
many
possible
choices
of
construction
data
for
the
extension.]
(iv)
Given
construction
data
“(k,
X,
Y
→
Y
,
Σ)”
or
“(k,
k,
X,
Y
→
Y
,
Σ)”
as
in
(i),
(ii),
(iii),
we
shall
refer
to
“k”
as
the
construction
data
field,
to
“X”
as
the
construction
data
base-stack
[or
base-scheme,
if
X
is
a
scheme],
to
“Y
”
as
the
con-
struction
data
covering,
to
“Y
”
as
the
construction
data
covering
compactification,
and
to
“Σ”
as
the
construction
data
prime
set.
Also,
we
shall
refer
to
a
portion
of
the
construction
data
“(k,
X,
Y
→
Y
,
Σ)”
or
“(k,
k,
X,
Y
→
Y
,
Σ)”
as
in
(i),
(ii),
(iii),
as
partial
construction
data.
If
every
prime
dividing
the
order
of
a
finite
quotient
group
of
Δ
is
invertible
in
k,
then
we
shall
refer
to
the
construction
data
in
question
as
base-prime.
The
following
result
is
well-known,
but
we
give
the
proof
below
for
lack
of
an
appropriate
reference
in
the
case
where
[in
the
notation
of
the
above
discussion]
X
is
not
necessarily
proper.
Proposition
2.2.
(Topological
Finite
Generation)
Any
profinite
group
Δ
of
GFG-type
is
topologically
finitely
generated.
Proof.
Write
(k,
X,
Y
→
Y
,
Σ)
for
a
choice
of
construction
data
for
Δ.
Since
a
profinite
fundamental
group
is
topologically
finitely
generated
if
and
only
if
it
admits
an
open
subgroup
that
is
topologically
finitely
generated,
we
may
assume
that
X
=
Y
;
moreover,
by
applying
de
Jong’s
theory
of
alterations
[as
reviewed,
for
instance,
in
Lemma
A.10
of
the
Appendix],
we
may
assume
that
Y
is
projective
def
and
k-smooth,
and
that
D
=
Y
\
Y
is
a
divisor
with
normal
crossings
on
Y
.
Since
we
are
only
concerned
with
Δ,
we
may
assume
that
k
is
algebraically
closed,
hence,
in
particular,
infinite.
Now
suppose
that
dim(Y
)
≥
2.
Then
since
Y
is
smooth
and
projective
[over
k],
it
follows
that
there
exists
a
connected,
k-smooth
closed
subscheme
C
⊆
Y
obtained
by
intersecting
Y
with
a
sufficiently
general
hyperplane
section
H
such
that
D
H
forms
a
divisor
with
normal
crossings
on
C.
Write
def
C
=
C
Y
(
=
∅).
Now
if
Z
→
Y
is
any
connected
finite
étale
covering
that
is
tamely
ramified
over
D,
then
write
Z
→
Y
for
the
normalization
of
Y
in
Z.
Thus,
since
Z
is
tamely
ramfied
over
D
—
so,
by
the
log
purity
theorem
reviewed
above,
one
may
apply
the
well-known
theory
of
log
étale
morphisms
to
describe
the
local
structure
of
Z
→
Y
—
and
D
intersects
C
transversely,
it
follows
immediately
def
that
Z
C
=
Z
×
Y
C
is
normal.
On
the
other
hand,
since
the
closed
subscheme
Z
C
⊆
Z
arises
as
the
zero
locus
of
a
nonzero
section
of
an
ample
line
bundle
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
I
19
on
the
normal
scheme
Z,
it
thus
follows
[cf.,
[SGA2],
XI,
3.11;
[SGA2],
XII,
2.4]
that
Z
C
is
connected,
hence
[since
Z
C
is
normal]
irreducible.
But
this
implies
def
that
Z
C
=
Z
×
Y
C
=
Z
C
Y
is
connected.
Moreover,
this
connectedness
of
Z
C
for
arbitrary
choices
of
the
covering
Z
→
Y
implies
that
the
natural
morphism
π
1
tame
(C)
→
π
1
tame
(Y
)
is
surjective.
Thus,
by
induction
on
dim(Y
),
it
suffices
to
prove
Proposition
2.2
in
the
case
where
Y
is
a
curve.
But
in
this
case,
[as
is
well-
known]
Proposition
2.2
follows
by
deforming
Y
→
Y
to
a
curve
in
characteristic
zero,
in
which
case
the
desired
topological
finite
generation
follows
from
the
well-
known
structure
of
the
topological
fundamental
group
of
a
Riemann
surface
of
finite
type.
Proposition
2.3.
(Slimness
and
Elasticity
for
Hyperbolic
Orbicurves)
(i)
Let
Δ
be
a
profinite
group
of
GFG-type
that
admits
partial
construction
data
(k,
X,
Σ)
[consisting
of
the
construction
data
field,
construction
data
base-
stack,
and
construction
data
prime
set]
such
that
X
is
a
hyperbolic
orbicurve
[cf.
§0],
and
Σ
contains
a
prime
invertible
in
k.
Then
Δ
is
slim
and
elastic.
(ii)
Let
1
→
Δ
→
Π
→
G
→
1
be
an
extension
of
GSAFG-type
that
admits
partial
construction
data
(k,
X,
Σ)
[consisting
of
the
construction
data
field,
construction
data
base-stack,
and
construction
data
prime
set]
such
that
X
is
a
hyperbolic
orbicurve,
Σ
=
∅,
and
k
is
either
an
MLF
or
an
NF.
Then
Π
is
slim,
but
not
elastic.
Proof.
Assertion
(i)
is
the
easily
verified
“generalization
to
orbicurves
over
fields
of
arbitrary
characteristic”
of
[MT],
Proposition
1.4;
[MT],
Theorem
1.5
[cf.
also
Proposition
1.3,
(i)].
The
slimness
portion
of
assertion
(ii)
follows
immediately
from
the
slimness
portion
of
assertion
(i),
together
with
the
slimness
portion
of
Theorem
1.7,
(ii),
(iii);
the
fact
that
Π
is
not
elastic
follows
from
the
existence
of
the
nontrivial,
topologically
finitely
generated
[cf.
Proposition
2.2],
closed,
normal,
infinite
index
subgroup
Δ
⊆
Π.
Definition
2.4.
For
i
=
1,
2,
let
1
→
Δ
i
→
Π
i
→
G
i
→
1
be
an
extension
which
is
either
of
AFG-type
or
of
GSAFG-type.
Suppose
that
φ
:
Π
1
→
Π
2
is
a
continuous
homomorphism
of
profinite
groups.
Then:
(i)
We
shall
say
that
φ
is
absolute
if
φ
is
open
[i.e.,
has
open
image].
(ii)
We
shall
say
that
φ
is
semi-absolute
(respectively,
pre-semi-absolute)
if
φ
is
absolute,
and,
moreover,
the
image
of
φ(Δ
1
)
in
G
2
is
trivial
(respectively,
either
trivial
or
of
infinite
index
in
G
2
).
20
SHINICHI
MOCHIZUKI
(iii)
We
shall
say
that
φ
is
strictly
semi-absolute
(respectively,
pre-strictly
semi-
absolute)
if
φ
is
semi-absolute,
and,
moreover,
the
subgroup
φ(Δ
1
)
⊆
Δ
2
is
open
(respectively,
either
open
or
nontrivial).
Proposition
2.5.
i
=
1,
2,
let
(First
Properties
of
Absolute
Homomorphisms)
For
1
→
Δ
i
→
Π
i
→
G
i
→
1
be
an
extension
which
is
either
of
AFG-type
or
of
GSAFG-type;
(k
i
,
X
i
,
Σ
i
)
partial
construction
data
for
Π
i
G
i
[consisting
of
the
construction
data
field,
construction
data
base-stack,
and
construction
data
prime
set].
Suppose
that
φ
:
Π
1
→
Π
2
is
a
continuous
homomorphism
of
profinite
groups.
Then:
(i)
The
following
implications
hold:
φ
strictly
semi-absolute
=⇒
φ
pre-strictly
semi-absolute
=⇒
φ
semi-absolute
=⇒
φ
pre-semi-absolute
=⇒
φ
absolute.
(ii)
Suppose
that
k
2
is
an
NF.
Then
“φ
semi-absolute”
⇐⇒
“φ
pre-semi-
absolute”
⇐⇒
“φ
absolute”.
(iii)
Suppose
that
k
2
is
an
MLF.
Then
“φ
semi-absolute”
⇐⇒
“φ
pre-semi-
absolute”.
(iv)
Suppose
that
k
1
either
an
FF
or
an
MLF;
that
X
2
is
a
hyperbolic
orbicurve;
and
that
Σ
2
is
of
cardinality
>
1.
Then
“φ
pre-strictly
semi-absolute”
⇐⇒
“φ
semi-absolute”.
(v)
Suppose
that
X
2
is
a
hyperbolic
orbicurve,
and
that
Σ
2
contains
a
prime
invertible
in
k
2
.
Then
“φ
strictly
semi-absolute”
⇐⇒
“φ
pre-strictly
semi-
absolute”.
Proof.
Assertion
(i)
follows
immediately
from
the
definitions.
Since
Δ
1
is
topo-
logically
finitely
generated
[cf.
Proposition
2.2],
assertion
(ii)
(respectively,
(iii))
follows
immediately,
in
light
of
assertion
(i),
from
the
fact
that
G
2
is
very
elastic
[cf.
Theorem
1.7,
(iii)]
(respectively,
elastic
[cf.
Theorem
1.7,
(ii)]).
To
verify
assertion
(iv),
it
suffices,
in
light
of
assertion
(i),
to
consider
the
case
where
φ
is
semi-absolute,
but
not
pre-strictly
semi-absolute.
Then
since
Δ
2
is
elastic
[cf.
the
hypothesis
on
Σ
2
;
Proposition
2.3,
(i)],
and
Δ
1
is
topologically
finitely
generated
[cf.
Proposition
2.2],
it
follows
that
the
subgroup
φ(Δ
1
)
⊆
Δ
2
is
either
open
or
trivial.
Since
φ
is
not
pre-strictly
semi-absolute,
we
thus
conclude
that
φ(Δ
1
)
=
{1},
so
φ
induces
an
open
homomorphism
G
1
→
Π
2
.
That
is
to
say,
every
sufficiently
small
open
subgroup
Δ
∗
2
⊆
Δ
2
admits
a
surjection
H
1
Δ
∗
2
for
some
closed
subgroup
H
1
⊆
G
1
.
On
the
other
hand,
since
X
2
is
a
hyperbolic
orbicurve,
and
Σ
2
is
of
cardinality
>
1,
it
follows
[e.g.,
from
the
well-known
structure
of
topological
fun-
damental
groups
of
hyperbolic
Riemann
surfaces
of
finite
type]
that
we
may
take
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
I
21
Δ
∗
2
such
that
Δ
∗
2
admits
quotients
Δ
∗
2
F
,
Δ
∗
2
F
,
where
F
(respectively,
F
)
is
a
nonabelian
free
pro-p
(respectively,
pro-p
)
group,
for
distinct
p
,
p
∈
Σ
2
.
But
this
contradicts
the
well-known
structure
of
G
1
,
when
k
1
is
either
an
FF
or
an
MLF
—
i.e.,
the
fact
that
G
1
,
hence
also
H
1
,
may
be
written
as
an
extension
of
a
meta-abelian
group
by
a
pro-p
subgroup,
for
some
prime
p.
[Here,
we
recall
that
this
fact
is
immediate
if
k
1
is
an
FF,
in
which
case
G
1
is
abelian,
and
follows,
for
instance,
from
[NSW],
Theorem
7.5.2;
[NSW],
Corollary
7.5.6,
(i),
when
k
1
is
a
MLF.]
Assertion
(v)
follows
immediately
from
the
elasticity
of
Δ
2
[cf.
Proposition
2.3,
(i)],
together
with
the
topological
finite
generation
of
Δ
1
[cf.
Proposition
2.2].
Theorem
2.6.
(Field
Types
and
Group-theoreticity)
Let
1
→
Δ
→
Π
→
G
→
1
be
an
extension
which
is
either
of
AFG-type
or
of
GSAFG-type;
(k,
X,
Σ)
partial
construction
data
[consisting
of
the
construction
data
field,
construction
data
base-stack,
and
construction
data
prime
set]
for
Π
G.
Suppose
further
that
k
is
either
an
FF,
an
MLF,
or
an
NF,
and
that
every
prime
∈
Σ
is
invertible
in
k.
If
H
is
a
profinite
group,
j
∈
{1,
2},
and
l
∈
Primes,
write
def
δ
l
j
(H)
=
dim
Q
l
(H
j
(H,
Q
l
))
∈
N
{∞}
def
jl
(Π)
=
sup
J⊆Π
{δ
l
j
(J)}
∈
N
{∞}
θ
j
(Π)
=
{l
|
jl
(Π)
≥
3
−
j}
⊆
Primes
def
[where
J
ranges
over
the
open
subgroups
of
Π];
also,
we
set
def
ζ(H)
=
sup
p,p
∈Primes
{δ
p
1
(H)
−
δ
p
1
(H)}
∈
Z
{∞}
whenever
δ
l
1
(H)
<
∞,
∀l
∈
Primes.
Then:
(i)
Suppose
that
k
is
an
FF.
Then
Π
is
topologically
finitely
generated;
the
natural
surjections
Π
ab-t
G
ab-t
;
G
G
ab-t
are
isomorphisms.
In
particular,
the
kernel
of
the
quotient
Π
G
may
be
char-
acterized
[“group-theoretically”]
as
the
kernel
of
the
quotient
Π
Π
ab-t
[cf.
[Tama1],
Proposition
3.3,
(ii),
in
the
case
of
curves].
Moreover,
for
every
open
subgroup
H
⊆
Π,
and
every
prime
number
l,
δ
l
1
(H)
=
1.
(ii)
Suppose
that
k
is
an
MLF
of
residue
characteristic
p.
Then
Π
is
topologi-
cally
finitely
generated;
in
particular,
for
every
open
subgroup
H
⊆
Π,
and
every
prime
number
l,
δ
l
1
(H)
is
finite.
Moreover,
δ
l
1
(G)
=
1
if
l
=
p,
δ
p
1
(G)
=
[k
:
Q
p
]+1;
the
quantity
δ
l
1
(Π)
−
δ
l
1
(G)
is
=
0
if
l
∈
/
Σ,
and
is
independent
of
l
if
l
∈
Σ.
Finally,
1
p
(Π)
=
∞;
in
particular,
the
cardinality
of
θ
1
(Π)
is
always
≥
1.
22
SHINICHI
MOCHIZUKI
(iii)
Let
k
be
as
in
(ii).
Then
θ
2
(Π)
⊆
Σ.
If,
moreover,
the
cardinality
of
θ
1
(Π)
is
≥
2,
then
θ
2
(Π)
=
Σ.
(iv)
Let
k
be
as
in
(ii).
Then
every
almost
pro-omissive
topologically
finitely
generated
closed
normal
subgroup
of
Π
is
contained
in
Δ.
If,
moreover,
Σ
=
Primes,
then
the
kernel
of
the
quotient
Π
G
may
be
characterized
[“group-
theoretically”]
as
the
maximal
almost
pro-omissive
topologically
finitely
gen-
erated
closed
normal
subgroup
of
Π.
(v)
Let
k
be
as
in
(ii).
If
θ
2
(Π)
=
Primes,
then
write
Θ
⊆
Π
for
the
maximal
almost
pro-omissive
topologically
finitely
generated
closed
nor-
mal
subgroup
of
Π,
whenever
a
unique
such
maximal
subgroup
exists;
if
θ
2
(Π)
=
def
Primes,
or
there
does
not
exist
a
unique
such
maximal
subgroup,
set
Θ
=
{1}
⊆
Π.
Then
def
ζ(Π)
=
ζ(Π/Θ)
=
[k
:
Q
p
]
[cf.
the
finiteness
portion
of
(ii)].
In
particular,
the
kernel
of
the
quotient
Π
G
may
be
characterized
[“group-theoretically”
—
since
“θ
2
(−)”,
“ζ(−)”,
“ζ(−)”
are
“group-theoretic”]
as
the
intersection
of
the
open
subgroups
H
⊆
Π
such
that
ζ(H)/ζ(Π)
=
[Π
:
H].
(vi)
Suppose
that
k
is
an
NF.
Then
the
natural
surjection
Π
ab-t
G
ab-t
is
an
isomorphism.
The
kernel
of
the
quotient
Π
G
may
be
characterized
[“group-
theoretically”]
as
the
maximal
topologically
finitely
generated
closed
normal
sub-
group
of
Π.
In
particular,
Π
is
not
topologically
finitely
generated.
Proof.
Write
X
→
A
for
the
Albanese
morphism
associated
to
X.
[We
refer
to
the
Appendix
for
a
review
of
the
theory
of
Albanese
varieties
—
cf.,
especially,
Corollary
A.11,
Remark
A.11.2.]
Thus,
A
is
a
torsor
over
a
semi-abelian
variety
over
k
such
that
the
morphism
X
→
A
induces
an
isomorphism
∼
Δ
ab-t
⊗
Z
l
→
T
l
(A)
onto
the
l-adic
Tate
module
T
l
(A)
of
A
for
all
l
∈
Σ.
Note,
moreover,
that
for
l
∈
Σ,
the
quotient
of
Δ
determined
by
the
image
of
Δ
in
the
pro-l
completion
of
Π
ab-t
factors
through
the
quotient
∼
Δ
Δ
ab-t
⊗
Z
l
→
T
l
(A)
T
l
(A)/G
—
where
we
use
the
notation
“/G”
to
denote
the
maximal
torsion-free
quotient
on
which
G
acts
trivially.
Next,
whenever
k
is
an
MLF,
let
us
write,
for
l
∈
Σ,
∼
def
def
Δ
ab-t
Δ
ab-t
⊗
Z
l
→
T
l
(A)
R
l
=
R
⊗
Z
l
Q
l
=
Q
⊗
Z
l
for
the
pro-l
portion
of
the
quotients
T
(A)
R
Q
of
Lemma
2.7,
(i),
(ii),
below
[in
which
we
take
“k”
to
be
k
and
“B”
to
be
the
semi-abelian
variety
over
which
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
I
23
A
is
a
torsor].
Here,
we
observe
that
Q
l
is
simply
the
quotient
T
l
(A)/G
considered
above.
Thus,
the
Z
l
-ranks
of
R
l
,
Q
l
are
independent
of
l
∈
Σ.
The
topological
finite
generation
portion
of
assertion
(i)
follows
immediately
together
with
the
topological
finite
generation
of
Δ
[cf.
from
the
fact
that
G
∼
=
Z,
Proposition
2.2].
The
remainder
of
assertion
(i)
follows
immediately
from
the
fact
that
T
l
(A)/G
=
0
[a
consequence
of
the
“Riemann
hypothesis
for
abelian
varieties
over
finite
fields”
—
cf.,
e.g.,
[Mumf],
p.
206].
In
a
similar
vein,
assertion
(vi)
follows
immediately
from
the
fact
that
T
l
(A)/G
=
0
[again
a
consequence
of
the
“Riemann
hypothesis
for
abelian
varieties
over
finite
fields”],
together
with
the
fact
that
G
is
very
elastic
[cf.
Theorem
1.7,
(iii)].
To
verify
assertion
(ii),
let
us
first
observe
that
the
topological
finite
generation
of
Π
follows
from
that
of
Δ
[cf.
Proposition
2.2],
together
with
that
of
G
[cf.
[NSW],
Theorem
7.5.10].
Next,
let
us
recall
the
well-known
fact
that
δ
l
1
(G)
=
1
if
l
=
p,
δ
p
1
(G)
=
[k
:
Q
p
]
+
1
[cf.
our
discussion
of
local
class
field
theory
in
the
proofs
of
Proposition
1.5;
Theorem
1.7,
(ii)];
in
particular,
ζ(G)
=
[k
:
Q
p
].
Moreover,
the
existence
of
a
rational
point
of
A
over
some
finite
extension
of
k
[which
determines
a
Galois
section
of
the
étale
fundamental
group
of
A
over
some
open
subgroup
of
G]
implies
that
δ
l
1
(Π)
=
δ
l
1
(G)
+
dim
Q
l
(Q
l
⊗
Q
l
)
[where
we
recall
that
dim
Q
l
(Q
l
⊗
Q
l
)
is
independent
of
l]
for
l
∈
Σ,
δ
l
1
(Π)
=
δ
l
1
(G)
for
l
∈
/
Σ.
Thus,
by
considering
extensions
of
k
of
arbitrarily
large
degree,
we
obtain
that
1
p
(Π)
=
∞.
This
completes
the
proof
of
assertion
(ii).
Next,
we
consider
assertion
(iii).
First,
let
us
consider
the
“E
2
-term”
of
the
Leray
spectral
sequence
of
the
group
extension
1
→
Δ
→
Π
→
G
→
1.
Since
G
is
of
cohomological
dimension
2
[cf.,
e.g.,
[NSW],
Theorem
7.1.8,
(i)],
and
δ
l
2
(G)
=
0
for
all
l
∈
Primes
[cf.,
e.g.,
[NSW],
Theorem
7.2.6],
the
spectral
sequence
yields
an
equality
δ
l
2
(Π)
=
0
if
l
∈
/
Σ,
and
a
pair
of
injections
H
1
(G,
Hom(R
l
,
Q
l
))
→
H
1
(G,
Hom(Δ
ab-t
,
Q
l
))
→
H
2
(Π,
Q
l
)
if
l
∈
Σ
[cf.
Lemma
2.7,
(iii),
below].
By
applying
the
analogue
of
this
conclusion
for
an
arbitrary
open
subgroup
H
⊆
Π,
we
thus
obtain
that
δ
l
2
(H)
=
0
if
l
∈
/
Σ,
2
2
i.e.,
that
l
(Π)
=
0
if
l
∈
/
Σ;
this
already
implies
that
if
l
∈
/
Σ,
then
l
∈
/
θ
(Π),
i.e.,
that
θ
2
(Π)
⊆
Σ.
If
the
cardinality
of
θ
1
(Π)
is
≥
2,
then
there
exists
some
open
subgroup
H
⊆
Π
and
some
l
∈
Primes
such
that
δ
l
1
(H)
≥
2,
l
=
p.
Now
we
may
assume
without
loss
of
generality
that
H
acts
trivially
on
the
quotient
R;
also
to
simplify
notation,
we
may
replace
Π
by
H
and
assume
that
H
=
Π.
Then
[since
δ
l
1
(G)
=
1,
by
assertion
(ii)]
the
fact
that
δ
l
1
(Π)
≥
2
implies
that
l
∈
Σ,
and
dim
Q
l
(R
l
⊗
Q
l
)
≥
1
[cf.
our
computation
in
the
proof
of
assertion
(ii)].
But
this
implies
that
for
any
l
∈
Σ,
we
have
dim
Q
l
(R
l
⊗
Q
l
)
≥
1,
hence
that
H
1
(G,
Hom(R
l
,
Q
l
))
=
H
1
(G,
Q
l
)
⊗
Hom(R
l
,
Q
l
)
=
0.
Thus,
by
the
injections
discussed
above,
we
conclude
that
2
l
(Π)
≥
δ
l
2
(Π)
≥
1,
so
l
∈
θ
2
(Π).
This
completes
the
proof
of
assertion
(iii).
24
SHINICHI
MOCHIZUKI
Assertion
(iv)
follows
immediately
from
the
existence
of
a
surjection
G
Z
[cf.,
e.g.,
Proposition
1.5,
(ii)],
together
with
the
elasticity
of
G
[cf.
Theorem
1.7,
(ii)],
and
the
topological
finite
generation
of
Δ
[cf.
Proposition
2.2].
Next,
we
consider
assertion
(v).
First,
let
us
observe
that
whenever
Σ
=
Primes,
it
follows
from
assertion
(ii)
that
ζ(Π)
=
ζ(G)
=
[k
:
Q
p
].
Now
we
consider
the
case
θ
2
(Π)
=
Primes.
In
this
case,
Θ
=
{1}
[by
def-
inition],
and
θ
2
(Π)
=
Σ
=
Primes
[by
assertion
(iii)].
Thus,
we
obtain
that
ζ(Π)
=
ζ(Π/Θ)
=
[k
:
Q
p
],
as
desired
[cf.
[Mzk6],
Lemma
1.1.4,
(ii)].
Next,
we
consider
the
case
θ
1
(Π)
=
{p}
[i.e.,
θ
1
(Π)
is
of
cardinality
≥
2
—
cf.
as-
sertion
(ii)],
θ
2
(Π)
=
Primes.
In
this
case,
by
assertion
(iii),
we
conclude
that
Σ
=
θ
2
(Π)
=
Primes.
Thus,
by
assertion
(iv),
Θ
=
Δ,
so
ζ(Π/Θ)
=
ζ(G)
=
[k
:
Q
p
],
as
desired.
Finally,
we
consider
the
case
θ
1
(Π)
=
{p}
[i.e.,
θ
1
(Π)
is
of
cardinality
one],
θ
(Π)
=
Primes.
If
Σ
=
Primes,
then
it
follows
from
the
definition
of
Θ,
together
with
assertion
(iv),
that
Θ
=
Δ,
hence
that
ζ(Π/Θ)
=
ζ(G)
=
[k
:
Q
p
],
as
desired.
If,
on
the
other
hand,
Σ
=
Primes,
then
since
θ
1
(Π)
=
{p},
it
follows
[cf.
the
computation
in
the
proof
of
assertion
(ii)]
that
dim
Q
l
(Q
l
⊗
Q
l
)
=
0
for
all
primes
l
=
p,
hence
that
dim
Q
p
(Q
p
⊗
Q
p
)
=
0;
but
this
implies
that
δ
l
1
(Π)
=
δ
l
1
(G)
for
all
l
∈
Primes.
Now
since
Θ
⊆
Δ
[by
assertion
(iv)],
it
follows
that
δ
l
1
(Π)
≥
δ
l
1
(Π/Θ)
≥
δ
l
1
(G)
for
all
l
∈
Primes,
so
we
obtain
that
δ
l
1
(Π)
=
δ
l
1
(Π/Θ)
=
δ
l
1
(G)
for
all
l
∈
Primes.
But
this
implies
that
ζ(Π)
=
ζ(Π/Θ)
=
ζ(G)
=
[k
:
Q
p
],
as
desired.
This
completes
the
proof
of
assertion
(v).
2
Remark
2.6.1.
When
[in
the
notation
of
Theorem
2.6]
X
is
a
smooth
proper
variety,
the
portion
of
Theorem
2.6,
(ii),
concerning
“δ
l
1
(Π)
−
δ
l
1
(G)”
is
essentially
equivalent
to
the
main
result
of
[Yoshi].
Lemma
2.7.
(Combinatorial
Quotients
of
Tate
Modules)
Suppose
that
k
is
an
MLF
[so
k
=
k].
Let
B
be
a
semi-abelian
variety
over
k.
Write
def
T
(B)
=
Hom(Q/Z,
B(k))
for
the
Tate
module
of
B.
Then:
(i)
The
maximal
torsion-free
quotient
module
T
(B)
Q
of
T
(B)
on
which
G
k
acts
trivially
is
a
finitely
generated
free
Z-module.
(ii)
There
exists
a
quotient
G
k
-module
T
(B)
R
such
that
the
following
properties
hold:
(a)
R
is
a
finitely
generated
free
Z-module;
(b)
the
action
of
G
k
on
R
factors
through
a
finite
quotient;
(c)
no
nonzero
torsion-free
subquotient
S
of
def
the
G
k
-module
N
=
Ker(T
(B)
R)
satisfies
the
property
that
the
resulting
action
of
G
k
on
S
factors
through
a
finite
quotient.
(iii)
If
R
is
as
in
(ii),
then
the
natural
map
→
H
1
(G
k
,
Hom(T
(B),
Z))
H
1
(G
k
,
Hom(R,
Z))
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
I
25
is
injective.
Proof.
Assertion
(i)
is
literally
the
content
of
[Mzk6],
Lemma
1.1.5.
Assertion
(ii)
follows
immediately
from
the
proof
of
[Mzk6],
Lemma
1.1.5
[more
precisely,
the
“combinatorial”
quotient
“T
com
”
of
loc.
cit.].
Assertion
(iii)
follows
by
considering
the
long
exact
cohomology
sequence
associated
to
the
short
exact
sequence
0
→
→
Hom(T
(B),
Z)
→
Hom(N,
Z)
→
0,
since
the
fact
that
N
has
no
Hom(R,
Z)
nonzero
torsion-free
subquotients
on
which
G
k
acts
through
a
finite
quotient
implies
=
0.
that
H
0
(G
k
,
Hom(N,
Z))
Corollary
2.8.
(Field
Types
and
Absolute
Homomorphisms)
For
i
=
1,
2,
let
1
→
Δ
i
→
Π
i
→
G
i
→
1,
k
i
,
X
i
,
Σ
i
,
φ
:
Π
1
→
Π
2
be
as
in
Proposition
2.5.
Suppose
further
that
k
i
is
either
an
FF,
an
MLF,
or
an
NF,
and
that
every
prime
∈
Σ
i
is
invertible
in
k
i
.
Then:
(i)
Suppose
further
that
φ
is
absolute.
Then
the
field
type
of
k
1
is
≥
[cf.
§0]
the
field
type
of
k
2
.
If,
moreover,
it
holds
either
that
both
k
1
and
k
2
are
FF’s
or
that
both
k
1
and
k
2
are
NF’s,
then
φ
is
semi-absolute,
i.e.,
φ(Δ
1
)
⊆
Δ
2
.
(ii)
Suppose
further
that
φ
is
an
isomorphism.
Then
the
field
types
of
k
1
,
k
2
coincide,
and
φ
is
strictly
semi-absolute,
i.e.,
φ(Δ
1
)
=
Δ
2
.
If,
moreover,
for
i
=
1,
2,
k
i
is
an
MLF
of
residue
characteristic
p
i
,
then
p
1
=
p
2
.
Proof.
Assertion
(i)
concerning
the
inequality
“≥”
follows
immediately
from
the
topological
finite
generation
portions
of
Theorem
2.6,
(i),
(ii),
(vi),
together
with
the
estimates
of
“δ
l
1
(−)”,
“
1
l
(−)”
in
Theorem
2.6,
(i),
(ii).
The
final
portion
of
assertion
follows,
in
the
case
of
FF’s,
from
Theorem
2.6,
(i),
and,
in
the
case
of
NF’s,
from
Proposition
2.5,
(ii).
Next,
we
consider
assertion
(ii).
The
fact
that
the
field
types
of
k
1
,
k
2
coincide
follows
from
assertion
(i)
applied
to
φ,
φ
−1
.
To
verify
that
φ
is
strictly
semi-absolute,
let
us
first
observe
that
every
semi-absolute
isomorphism
whose
inverse
is
also
semi-absolute
is
necessarily
strictly
semi-absolute.
Thus,
since
the
inverse
to
φ
satisfies
the
same
hypotheses
as
φ,
to
complete
the
proof
of
Corollary
2.8,
it
suffices
to
verify
that
φ
is
semi-absolute.
If
k
1
,
k
2
are
FF’s
(respectively,
MLF’s;
NF’s),
then
this
follows
immediately
from
the
“group-
theoretic”
characterizations
of
Π
i
G
i
in
Theorem
2.6,
(i)
(respectively,
Theorem
2.6,
(v);
Theorem
2.6,
(vi)).
Finally,
if,
for
i
=
1,
2,
k
i
is
an
MLF
of
residue
∼
characteristic
p
i
,
then
since
φ
induces
an
isomorphism
G
1
→
G
2
,
the
fact
that
p
1
=
p
2
follows,
for
instance,
from
[Mzk6],
Proposition
1.2.1,
(i).
Remark
2.8.1.
In
the
situation
of
Corollary
2.8,
suppose
further
that
k
2
is
an
MLF
of
residue
characteristic
p
2
;
that
X
2
is
a
hyperbolic
orbicurve;
that
Σ
2
⊆
{p
2
}
[cf.
Proposition
2.5,
(iv)];
and
that
if
Σ
2
=
∅,
then
k
1
is
an
NF.
Then
it
is
not
clear
to
the
author
at
the
time
of
writing
[but
of
interest
in
the
context
of
the
theory
of
the
present
§2!]
whether
or
not
there
exists
a
continuous
surjective
homomorphism
G
1
Π
2
[in
which
case,
by
Corollary
2.8,
(i),
k
1
is
either
an
NF
or
an
MLF].
26
SHINICHI
MOCHIZUKI
The
general
theory
discussed
so
far
for
arbitrary
X
becomes
substantially
sim-
pler
and
more
explicit,
when
X
is
a
hyperbolic
orbicurve.
Definition
2.9.
Let
G
be
a
profinite
group.
Then
we
shall
refer
to
as
an
aug-free
decomposition
of
G
any
pair
of
closed
subgroups
H
1
,
H
2
⊆
G
that
determine
an
isomorphism
of
profinite
groups
∼
H
1
×
H
2
→
G
such
that
H
1
is
a
slim,
topologically
finitely
generated,
augmented
pro-prime
[cf.
Definition
1.1,
(iii)]
profinite
group,
and
H
2
is
either
trivial
or
a
nonabelian
pro-Σ-
solvable
free
group
for
some
set
Σ
⊆
Primes
of
cardinality
≥
2.
In
this
situation,
we
shall
refer
to
H
1
as
the
augmented
subgroup
of
this
aug-free
decomposition
and
to
H
2
as
the
free
subgroup
of
this
aug-free
decomposition.
If
G
admits
an
aug-free
decomposition,
then
we
shall
say
that
G
is
of
aug-free
type.
If
G
is
of
aug-free
type,
with
nontrivial
free
subgroup,
then
we
shall
say
that
G
is
of
strictly
aug-free
type.
Proposition
2.10.
(First
Properties
of
Aug-free
Decompositions)
Let
∼
H
1
×
H
2
→
G
be
an
aug-free
decomposition
of
a
profinite
group
G,
in
which
H
1
is
the
aug-
mented
subgroup,
and
H
2
is
the
free
subgroup.
Then:
(i)
Let
J
be
a
topologically
finitely
generated,
augmented
pro-prime
group;
φ
:
J
→
G
a
continuous
homomorphism
of
profinite
groups
such
that
φ(J)
is
normal
in
some
open
subgroup
of
G.
Then
φ(J)
⊆
H
1
.
∼
(ii)
Aug-free
decompositions
are
unique
—
i.e.,
if
J
1
×
J
2
→
G
is
any
aug-
free
decomposition
of
G,
in
which
J
1
is
the
augmented
subgroup,
and
J
2
is
the
free
subgroup,
then
J
1
=
H
1
,
J
2
=
H
2
.
Proof.
First,
we
consider
assertion
(i).
Suppose
that
φ(J)
is
not
contained
in
H
1
.
Then
the
image
I
⊆
H
2
of
φ(J)
via
the
projection
to
H
2
is
a
nontrivial,
topologically
finitely
generated
closed
subgroup
which
is
normal
in
an
open
subgroup
of
H
2
.
Since
H
2
is
elastic
[cf.
[MT],
Theorem
1.5],
it
follows
that
I
is
open
in
H
2
,
hence
that
I
is
a
nonabelian
pro-Σ-solvable
free
group
for
some
set
Σ
⊆
Primes
of
cardinality
≥
2.
On
the
other
hand,
since
I
is
a
quotient
of
the
augmented
pro-prime
group
J,
it
follows
that
there
exists
a
p
∈
Primes
such
that
the
maximal
pro-(
=
p)
quotient
of
I
is
abelian.
But
this
implies
that
Σ
⊆
{p},
a
contradiction.
Next,
we
consider
assertion
(ii).
By
assertion
(i),
J
1
⊆
H
1
,
H
1
⊆
J
1
.
Thus,
H
1
=
J
1
.
Now
since
H
1
=
J
1
is
slim,
it
follows
that
the
centralizer
Z
H
1
(G)
(respectively,
Z
J
1
(G))
is
equal
to
H
2
(respectively,
J
2
),
so
H
2
=
J
2
,
as
desired.
Theorem
2.11.
curves)
Let
(Maximal
Pro-RTF-quotients
for
Hyperbolic
Orbi-
1
→
Δ
→
Π
→
G
→
1
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
I
27
be
an
extension
of
AFG-type;
(k,
X,
Σ)
partial
construction
data
[consisting
of
the
construction
data
field,
construction
data
base-stack,
and
construction
data
prime
set]
for
Π
G.
Suppose
that
k
is
an
MLF
of
residue
characteristic
p;
X
is
a
hyperbolic
orbicurve;
Σ
=
∅.
For
l
∈
Primes,
write
Π[l]
⊆
Π
for
the
maximal
almost
pro-l
topologically
finitely
generated
closed
normal
sub-
group
of
Π,
whenever
a
unique
such
maximal
subgroup
exists;
if
there
does
not
exist
def
a
unique
such
maximal
subgroup,
then
set
Π[l]
=
{1}.
In
the
following,
we
shall
use
a
subscript
“G”
to
denote
the
quotient
of
a
closed
subgroup
of
Π
determined
by
the
quotient
Π
G;
we
shall
use
the
super-
script
“RTF”
to
denote
the
maximal
pro-RTF-quotient
and
the
superscripts
“RTF-aug”,
“RTF-free”
to
denote
the
augmented
and
free
subgroups
of
the
max-
imal
pro-RTF-quotient
whenever
this
maximal
pro-RTF-quotient
is
of
aug-free
type.
Then:
(i)
Suppose
that
Π[l]
=
{1}
for
some
l
∈
Primes.
Then
Π[l]
=
Δ,
Σ
=
{l};
Π[l
]
=
{1}
for
all
l
∈
Primes
such
that
l
=
l.
(ii)
Suppose
that
Π[l]
=
{1}
for
all
l
∈
Primes.
Then
Σ
is
of
cardinality
≥
2.
Moreover,
for
every
open
subgroup
J
⊆
Π,
there
exists
an
open
subgroup
H
⊆
J
which
is
characteristic
as
a
subgroup
of
Π
such
that
H
RTF
is
of
aug-free
type.
In
particular,
[cf.
Proposition
2.10,
(ii)]
the
subquotients
H
RTF-aug
,
H
RTF-free
of
Π
are
characteristic.
(iii)
Suppose
that
Π[l]
=
{1}
for
all
l
∈
Primes.
Suppose,
moreover,
that
H
⊆
Π
is
an
open
subgroup
that
corresponds
to
a
finite
étale
covering
Z
→
X,
where
Z
is
a
hyperbolic
curve,
defined
over
a
finite
extension
k
Z
of
k
such
that
Z
has
stable
reduction
[cf.
§0]
over
the
ring
of
integers
O
k
Z
of
k
Z
;
that
Z(k
Z
)
=
∅;
that
the
dual
graph
Γ
Z
of
the
geometric
special
fiber
of
the
resulting
model
[cf.
§0]
over
O
k
Z
has
either
trivial
or
nonabelian
topological
fundamental
group;
and
that
the
Galois
action
of
G
on
Γ
Z
is
trivial.
Thus,
the
finite
Galois
coverings
of
the
graph
Γ
Z
of
degree
a
product
of
primes
∈
Σ
determine
a
pro-Σ
com
“combinatorial”
quotient
H
Δ
com
Δ
com-sol
for
the
maximal
H
;
write
Δ
H
H
com
pro-solvable
quotient
of
Δ
H
.
Then
the
quotient
RTF
H
H
G
×
Δ
com-sol
H
may
be
identified
with
the
maximal
pro-RTF-quotient
H
H
RTF
of
H;
more-
over,
this
product
decomposition
determines
an
aug-free
decomposition
of
H
RTF
.
Finally,
for
any
open
subgroup
J
⊆
Π,
there
exists
an
open
subgroup
H
⊆
J
which
is
characteristic
as
a
subgroup
of
Π
and,
moreover,
satisfies
the
above
hypotheses
on
“H”.
(iv)
Suppose
that
Π[l]
=
{1}
for
all
l
∈
Primes.
Let
H
⊆
J
⊆
Π
be
open
subgroups
of
Π
such
that
H
RTF
,
J
RTF
are
of
aug-free
type.
Then
we
have
iso-
morphisms
∼
RTF
J
RTF-aug
→
J
G
;
∼
RTF
J
RTF-free
→
Ker(J
RTF
J
G
)
28
SHINICHI
MOCHIZUKI
[arising
from
the
natural
morphisms
involved];
the
open
homomorphism
H
RTF
→
J
RTF
induced
by
φ
maps
H
RTF-aug
(respectively,
H
RTF-free
)
onto
an
open
subgroup
of
J
RTF-aug
(respectively,
J
RTF-free
).
Proof.
Since
Δ
is
elastic
[cf.
Proposition
2.3,
(i)],
every
nontrivial
topologically
finitely
generated
closed
normal
subgroup
of
Δ
is
open,
hence
almost
pro-Σ
for
Σ
⊆
Primes
if
and
only
if
Σ
⊇
Σ.
Also,
let
us
observe
that
by
Theorem
2.6,
(iv),
Π[l]
⊆
Δ
for
all
l
∈
Primes.
Thus,
if
Π[l]
=
{1}
for
some
l
∈
Primes,
then
it
follows
that
Σ
=
{l},
Π[l]
=
Δ,
and
that
Π[l
]
is
finite,
hence
trivial
[since
Δ
is
slim
—
cf.
Proposition
2.3,
(i)]
for
primes
l
=
l.
Also,
we
observe
that
if
Σ
is
of
cardinality
one,
i.e.,
Σ
=
{l}
for
some
l
∈
Primes,
then
Δ
=
Π[l]
=
{1}
[cf.
Theorem
2.6,
(iv)].
This
completes
the
proof
of
assertion
(i),
as
well
as
of
the
portion
of
assertion
(ii)
concerning
Σ.
Also,
we
observe
that
the
remainder
of
assertion
(ii)
follows
immediately
from
assertion
(iii).
Next,
we
consider
assertion
(iii).
Suppose
that
H
⊆
Π
satisfies
the
hypotheses
def
given
in
the
statement
of
assertion
(iii);
write
Δ
H
=
Δ
H.
Thus,
one
has
the
com
quotient
H
Δ
com
is
either
trivial
or
a
nonabelian
pro-Σ
free
group,
H
,
where
Δ
H
and
Σ
is
of
cardinality
≥
2
[cf.
the
portion
of
assertion
(ii)
concerning
Σ].
Write
ab-t
Δ
ab
R
for
the
maximal
pro-Σ
quotient
of
the
quotient
“R”
of
Lemma
H
=
Δ
H
2.7,
(ii),
associated
to
the
Albanese
variety
of
Z.
Now
I
claim
that
the
quotient
Δ
H
R
coincides
with
the
quotient
Δ
H
com
ab
(Δ
H
)
.
First,
let
us
observe
that
by
the
definition
of
R
[cf.
Lemma
2.7,
(ii)],
it
ab
follows
that
the
quotient
Δ
H
(Δ
com
factors
through
the
quotient
Δ
H
R.
H
)
ab
In
particular,
since,
for
l
∈
Σ,
the
modules
R⊗Z
l
,
(Δ
com
H
)
⊗Z
l
are
Z
l
-free
modules
of
rank
independent
of
l
∈
Σ
[cf.
Lemma
2.7,
(ii);
the
fact
that
Δ
com
is
pro-Σ
free],
H
it
suffices
to
show
that
these
two
ranks
are
equal,
for
some
l
∈
Σ.
Moreover,
let
us
observe
that
for
the
purpose
of
verifying
this
claim,
we
may
enlarge
Σ.
Thus,
it
suffices
to
show
that
the
two
ranks
are
equal
for
some
l
∈
Σ
such
that
l
=
p.
But
then
the
claim
follows
immediately
from
the
[well-known]
fact
that
by
the
“Riemann
hypothesis
for
abelian
varieties
over
finite
fields”
[cf.,
e.g.,
[Mumf],
p.
206],
all
powers
of
the
Frobenius
element
in
the
absolute
Galois
group
of
the
residue
field
of
k
act
with
eigenvalues
=
1
on
the
pro-l
abelianizations
of
the
fundamental
groups
of
the
geometric
irreducible
components
of
the
smooth
locus
of
the
special
fiber
of
the
stable
model
of
Z
over
O
k
Z
.
This
completes
the
proof
of
the
claim.
Now
let
us
write
H
H
com
for
the
quotient
of
H
by
Ker(Δ
H
Δ
com
H
).
Then
by
applying
the
above
claim
to
various
open
subgroups
of
H,
we
conclude
that
the
quotient
H
H
RTF
factors
through
the
quotient
H
H
com
[i.e.,
we
have
a
∼
natural
isomorphism
H
RTF
→
(H
com
)
RTF
].
On
the
other
hand,
since
Z(k
Z
)
=
∅,
it
follows
that
H
H
G
,
hence
also
H
com
H
G
admits
a
section
s
:
H
G
→
H
com
whose
image
lies
in
the
kernel
of
the
quotient
H
com
Δ
com
[cf.
the
proof
of
H
[Mzk3],
Lemma
1.4].
In
particular,
we
conclude
that
the
conjugation
action
of
H
G
∼
on
Δ
com
=
Ker(H
com
H
G
)
⊆
H
com
arising
from
s
is
trivial.
Thus,
s
determines
H
a
direct
product
decomposition
∼
H
com
→
H
G
×
Δ
com
H
∼
∼
RTF
RTF
—
hence
a
direct
product
decomposition
H
RTF
→
(H
com
)
RTF
→
H
G
×(Δ
com
.
H
)
com
Moreover,
since
Δ
H
is
either
trivial
or
nonabelian
pro-Σ
free,
it
follows
immedi-
RTF
ately
that
the
quotient
Δ
com
(Δ
com
may
be
identified
with
the
quotient
H
H
)
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
I
29
Δ
com
Δ
com-sol
,
where
Δ
com-sol
is
either
trivial
or
nonabelian
pro-Σ-solvable
free.
H
H
H
RTF
Since
H
G
is
slim,
augmented
pro-prime,
and
topologically
finitely
generated
[cf.
Proposition
1.5,
(i),
(ii);
Theorem
2.6,
(ii)],
we
thus
conclude
that
we
have
obtained
an
aug-free
decomposition
of
H
RTF
,
as
asserted
in
the
statement
of
assertion
(iii).
Finally,
given
an
open
subgroup
J
⊆
Π,
the
existence
of
an
open
subgroup
H
⊆
J
which
satisfies
the
hypotheses
on
“H”
in
the
statement
of
assertion
(iii)
follows
immediately
from
well-known
facts
concerning
stable
curves
over
discretely
valued
fields
[cf.,
e.g.,
the
“stable
reduction
theorem”
of
[DM];
the
fact
that
Σ
=
∅,
so
that
one
may
assume
that
Γ
Z
is
as
large
as
one
wishes
by
passing
to
admissible
coverings].
The
fact
that
one
can
choose
H
to
be
characteristic
follows
immediately
from
the
characteristic
nature
of
Δ
[cf.,
e.g.,
Corollary
2.8,
(ii)],
together
with
the
fact
that
Δ,
Π
are
topologically
finitely
generated
[cf.,
e.g.
Proposition
2.2;
Theorem
2.6,
(ii)].
This
completes
the
proof
of
assertion
(iii).
Finally,
we
consider
assertion
(iv).
First,
we
observe
that
since
the
augmented
and
free
subgroups
of
any
aug-free
decomposition
are
slim
[cf.
Definition
2.9;
[MT],
Proposition
1.4],
hence,
in
particular,
do
not
contain
any
nontrivial
closed
normal
fi-
nite
subgroups,
we
may
always
replace
H
by
an
open
subgroup
of
H
that
satisfies
the
same
hypotheses
as
H.
In
particular,
we
may
assume
that
H
is
an
open
subgroup
“H”
as
in
assertion
(iii)
[which
exists,
by
assertion
(iii)].
Then
by
Proposition
2.10,
(i),
the
image
of
H
RTF-aug
in
J
RTF
is
contained
in
J
RTF-aug
,
so
we
obtain
a
mor-
RTF
phism
H
RTF-aug
→
J
RTF-aug
.
By
assertion
(iii),
H
RTF-free
=
Ker(H
RTF
H
G
),
RTF-aug
RTF
and
the
natural
morphism
H
→
H
G
is
an
isomorphism.
Since
H
G
→
J
G
,
RTF
RTF
hence
also
H
G
→
J
G
,
is
clearly
an
open
homomorphism,
we
thus
conclude
RTF
that
the
natural
morphism
H
RTF-aug
→
J
G
,
hence
also
the
natural
morphism
RTF-aug
RTF
RTF
J
→
J
G
,
is
open.
Thus,
the
image
of
J
RTF-free
in
J
G
commutes
with
an
RTF
RTF-aug
RTF
open
subgroup
of
J
G
[i.e.,
the
image
of
J
in
J
G
],
so
by
the
slimness
of
RTF
RTF-free
RTF
J
G
[cf.
Proposition
1.5,
(i)],
we
conclude
that
J
⊆
Ker(J
RTF
J
G
).
RTF-aug
RTF
In
particular,
we
obtain
a
surjection
J
J
G
,
hence
an
exact
sequence
RTF
→
1
1
→
N
→
J
RTF-aug
→
J
G
def
RTF
—
where
we
write
N
=
Ker(J
RTF-aug
J
G
)
⊆
J
RTF-aug
⊆
J
RTF
.
Note,
RTF
moreover,
that
since
J
G
is
an
augmented
pro-p
group
[cf.
Proposition
1.5,
(ii)]
RTF
which
admits
a
surjection
J
G
Z
p
×
Z
p
[cf.
the
computation
of
“δ
p
1
(−)”
in
Theorem
2.6,
(ii)],
it
follows
immediately
that
[the
augmented
pro-prime
group]
RTF
J
RTF-aug
is
an
augmented
pro-p
group
whose
augmentation
factors
through
J
G
;
in
particular,
we
conclude
that
N
is
pro-p.
Also,
we
observe
that
since
the
composite
RTF
RTF
H
RTF-free
→
H
G
→
J
G
is
trivial,
it
follows
that
the
projection
under
the
quotient
J
RTF
J
RTF-aug
of
the
image
of
H
RTF-free
in
J
RTF
is
contained
in
N
.
Now
I
claim
that
to
complete
the
proof
of
assertion
(iv),
it
suffices
to
verify
that
N
=
{1}
[or,
equivalently,
since
J
RTF-aug
is
slim,
that
N
is
finite].
Indeed,
∼
RTF
,
if
N
=
{1},
then
we
obtain
immediately
the
isomorphisms
J
RTF-aug
→
J
G
∼
RTF-free
RTF
RTF
J
→
Ker(J
J
G
).
Moreover,
by
the
above
discussion,
if
N
=
{1},
then
it
follows
that
the
image
of
H
RTF-free
in
J
RTF
is
contained
in
J
RTF-free
.
Since
the
homomorphism
H
RTF
→
J
RTF
is
open,
this
implies
that
the
open
homomor-
phism
H
RTF
J
RTF
induced
by
φ
maps
H
RTF-aug
(respectively,
H
RTF-free
)
onto
an
open
subgroup
of
J
RTF-aug
(respectively,
J
RTF-free
),
as
desired.
This
completes
the
proof
of
the
claim.
30
SHINICHI
MOCHIZUKI
Next,
let
J
⊆
J
be
an
open
subgroup
that
arises
as
the
inverse
image
in
J
of
an
[open]
RTF-subgroup
J
G
⊆
J
G
[so
the
notation
“J
G
”
does
not
lead
to
any
contradictions].
Then
one
verifies
immediately
from
the
definitions
that
any
RTF-
subgroup
of
J
G
(respectively,
J)
determines
an
RTF-subgroup
of
J
G
(respectively,
J).
Thus,
the
natural
morphisms
RTF
J
RTF
→
J
G
;
G
J
RTF
→
J
RTF
are
injective.
Moreover,
the
subgroups
J
RTF-aug
J
RTF
,
J
RTF-free
of
J
RTF
clearly
determine
an
aug-free
decomposition
of
J
RTF
.
Thus,
from
the
point
of
view
of
verifying
the
finiteness
of
N
,
we
may
replace
J
by
J
[and
H
by
an
appropriate
smaller
open
subgroup
contained
in
J
and
satisfying
the
hypotheses
of
the
“H”
of
(iii)].
In
particular,
since
—
by
the
definition
of
“RTF”
and
of
the
subgroup
N
!
—
there
exists
a
J
such
that
N
⊆
J
RTF-aug
has
nontrivial
image
in
(J
RTF-aug
)
ab-t
,
we
may
assume
without
loss
of
generality
that
N
has
nontrivial
image
in
(J
RTF-aug
)
ab-t
.
Thus,
we
have
RTF
(δ
p
1
(J)
≥)
δ
p
1
(J
RTF-aug
)
>
δ
p
1
(J
G
)
=
δ
p
1
(J
G
)
def
RTF
[cf.
the
notation
of
Theorem
2.6],
i.e.,
s
J
=
δ
p
1
(J
RTF-aug
)
−
δ
p
1
(J
G
)
>
0.
By
Theorem
2.6,
(ii),
this
already
implies
that
p
∈
Σ.
In
a
similar
vein,
let
J
⊆
J
be
an
open
subgroup
that
arises
as
the
inverse
image
in
J
of
an
[open]
RTF-subgroup
J
RTF-free
⊆
J
RTF-free
.
Then
one
verifies
immediately
from
the
definitions
that
any
RTF-subgroup
of
J
determines
an
RTF-
subgroup
of
J.
Thus,
the
natural
morphism
J
RTF
→
J
RTF
is
injective,
with
image
equal
to
J
RTF-aug
×
J
RTF-free
.
Moreover,
the
subgroups
J
RTF-aug
,
J
RTF-free
of
J
RTF
clearly
determine
an
aug-free
decomposition
of
J
RTF
[so
the
notation
“J
RTF-free
”
does
not
lead
to
any
contradictions].
Since
[by
the
above
discussion
applied
to
J
instead
of
J]
J
RTF-free
maps
to
the
identity
in
J
RTF
G
,
we
thus
obtain
a
quotient
RTF
J
,
hence
a
quotient
J
RTF
J
RTF-aug
J
RTF
J
RTF
J
RTF-aug
=
J
RTF-aug
G
G
in
which
the
image
of
J
Δ
is
a
finite
normal
closed
subgroup,
hence
trivial
[since
is
slim
—
cf.
Proposition
1.5,
(i)].
That
is
to
say,
the
surjection
J
J
RTF
G
RTF-aug
J
RTF
to
the
pro-RTF-group
J
RTF
factors
through
J
G
,
hence
through
J
G
G
RTF
RTF
RTF
.
Thus,
we
obtain
a
surjection
J
G
J
RTF
whose
composite
J
G
J
G
G
RTF
RTF
J
G
→
J
G
with
the
natural
morphism
induced
by
the
inclusion
J
→
J
is
the
RTF
identity
[since
J
G
is
slim
[cf.
Proposition
1.5,
(i)],
and
all
of
these
maps
“lie
RTF
→
J
G
is
an
under
a
fixed
J”].
But
this
implies
that
the
natural
morphism
J
RTF
G
RTF-aug
isomorphism.
In
particular,
we
have
an
isomorphism
of
kernels
Ker(J
∼
RTF-aug
RTF
)
→
Ker(J
J
).
Thus,
from
the
point
of
view
of
verifying
the
J
RTF
G
G
finiteness
of
N
,
we
may
replace
J
by
J
[and
H
by
an
appropriate
smaller
open
subgroup
contained
in
J
and
satisfying
the
hypotheses
of
the
“H”
of
(iii)].
In
∼
particular,
since
J
RTF-aug
→
J
RTF-aug
,
we
may
assume
without
loss
of
generality
that
the
rank
r
J
of
the
pro-Σ
J
-solvable
free
group
J
RTF-free
[for
some
subset
Σ
J
⊆
Primes
of
cardinality
≥
2]
is
either
0
or
>
δ
p
1
(J
RTF-aug
).
In
particular,
if
l
∈
Σ
J
,
then
either
r
J
=
0
or
r
J
=
δ
l
1
(J
RTF-free
)
>
δ
p
1
(J
RTF-aug
)
≥
s
J
.
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
I
31
Now
we
compute:
Since
Σ
is
of
cardinality
≥
2,
let
l
∈
Σ
be
a
prime
=
p.
Then:
RTF
δ
l
1
(J
RTF-free
)
=
δ
l
1
(J
RTF-free
)
+
δ
l
1
(J
RTF-aug
)
−
δ
l
1
(J
G
)
RTF
)
=
δ
l
1
(J)
−
δ
l
1
(J
G
)
=
δ
l
1
(J
RTF
)
−
δ
l
1
(J
G
RTF
)
=
δ
p
1
(J)
−
δ
p
1
(J
G
)
=
δ
p
1
(J
RTF
)
−
δ
p
1
(J
G
RTF
)
=
δ
p
1
(J
RTF-free
)
+
s
J
=
δ
p
1
(J
RTF-free
)
+
δ
p
1
(J
RTF-aug
)
−
δ
p
1
(J
G
—
where
we
apply
the
“independence
of
l”
of
Theorem
2.6,
(ii).
Thus,
we
conclude
that
s
J
=
δ
l
1
(J
RTF-free
)
−
δ
p
1
(J
RTF-free
)
—
where
δ
l
1
(J
RTF-free
),
δ
p
1
(J
RTF-free
)
∈
{0,
r
J
}
[depending
on
whether
or
not
l,
p
belong
to
Σ
J
]
—
is
a
positive
integer.
But
this
implies
that
0
<
s
J
∈
{0,
r
J
,
−r
J
},
hence
that
s
J
=
r
J
>
0
—
in
contradiction
to
the
inequality
s
J
<
r
J
[which
holds
if
r
J
>
0].
This
completes
the
proof
of
assertion
(iv).
Remark
2.11.1.
One
way
of
thinking
about
the
content
of
Theorem
2.11,
(iv),
is
that
it
asserts
that
“aug-free
decompositions
of
maximal
pro-RTF-
quotients
play
an
analogous
[though
somewhat
more
complicated]
role
for
absolute
Galois
groups
of
MLF’s
to
the
role
played
by
torsion-free
abelianizations
for
absolute
Galois
groups
of
FF’s”
[cf.
Theorem
2.6,
(i)].
Corollary
2.12.
(Group-theoretic
Semi-absoluteness
via
Maximal
Pro-
RTF-quotients)
For
i
=
1,
2,
let
1
→
Δ
i
→
Π
i
→
G
i
→
1,
k
i
,
X
i
,
Σ
i
,
φ
:
Π
1
→
Π
2
be
as
in
Proposition
2.5.
Suppose
further
that
k
i
is
an
MLF;
X
i
is
a
hyperbolic
orbicurve;
Σ
i
=
∅.
Also,
for
i
=
1,
2,
let
us
write
Θ
i
⊆
Π
i
for
the
maximal
almost
pro-prime
topologically
finitely
generated
closed
normal
subgroup
of
Π
i
,
whenever
a
unique
such
maximal
subgroup
exists;
if
there
does
not
def
exist
a
unique
such
maximal
subgroup,
then
we
set
Θ
i
=
{1}.
Suppose
that
φ
is
absolute.
Then:
(i)
For
i
=
1,
2,
Θ
i
⊆
Δ
i
;
Θ
i
=
{1}
if
and
only
if
Σ
i
is
of
cardinality
one;
if
Θ
i
=
{1},
then
Θ
i
=
Δ
i
.
Finally,
φ(Θ
1
)
⊆
Θ
2
[so
φ
induces
a
morphism
Π
1
/Θ
1
→
Π
2
/Θ
2
].
(ii)
In
the
notation
of
Theorem
2.11,
φ
is
semi-absolute
[or,
equivalently,
pre-semi-absolute
—
cf.
Proposition
2.5,
(iii)]
if
and
only
if
the
following
[“group-
theoretic”]
condition
holds:
(∗
s-ab
)
For
i
=
1,
2,
let
H
i
⊆
Π
i
/Θ
i
be
an
open
subgroup
such
that
H
i
RTF
is
of
aug-free
type,
and
[the
morphism
induced
by]
φ
maps
H
1
into
H
2
.
Then
the
open
homomorphism
H
1
RTF
→
H
2
RTF
induced
by
φ
maps
H
1
RTF-free
into
H
2
RTF-free
.
32
SHINICHI
MOCHIZUKI
(iii)
If,
moreover,
Σ
2
is
of
cardinality
≥
2,
then
φ
is
semi-absolute
if
and
only
if
it
is
strictly
semi-absolute
[or,
equivalently,
pre-strictly
semi-absolute
—
cf.
Proposition
2.5,
(v)].
Proof.
First,
we
consider
assertion
(i).
By
Theorem
2.6,
(iv),
any
almost
pro-
prime
topologically
finitely
generated
closed
normal
subgroup
of
Π
i
—
hence,
in
particular,
Θ
i
—
is
contained
in
Δ
i
.
Thus,
by
Theorem
2.11,
(i),
(ii),
Θ
i
=
{1}
if
and
only
if
Σ
i
is
of
cardinality
one;
if
Θ
i
=
{1},
then
Θ
i
=
Δ
i
.
Now
to
show
that
φ(Θ
1
)
⊆
Θ
2
,
it
suffices
to
consider
the
case
where
φ(Θ
1
)
=
{1}
[so
Σ
1
is
of
cardinality
one].
Since
φ
is
absolute,
it
follows
that
φ(Θ
1
)
is
normal
in
some
open
subgroup
of
Π
2
.
Thus,
by
Theorem
2.6,
(iv),
we
have
φ(Θ
1
)
⊆
Δ
2
,
so
we
may
assume
that
Θ
2
=
{1}
[which
implies
that
Σ
2
is
of
cardinality
≥
2].
But
then
the
elasticity
of
Δ
2
[cf.
Proposition
2.3,
(i)]
implies
that
φ(Θ
1
)
is
an
open
subgroup
of
Δ
2
,
hence
that
φ(Θ
1
)
is
almost
pro-Σ
2
[for
some
Σ
2
of
cardinality
≥
2],
which
contradicts
the
fact
that
φ(Θ
1
)
is
almost
pro-Σ
1
[for
some
Σ
1
of
cardinality
one].
This
completes
the
proof
of
assertion
(i).
Next,
we
consider
assertion
(ii).
By
Proposition
2.5,
(iii),
one
may
replace
the
term
“semi-absolute”
in
assertion
(ii)
by
the
term
“pre-semi-absolute”.
By
assertion
(i),
for
i
=
1,
2,
either
Θ
i
=
{1}
or
Θ
i
=
Δ
i
;
in
either
case,
it
follows
from
Theorem
2.11,
(iv)
[cf.
also
Proposition
1.5,
(i),
(ii)],
that
[in
the
notation
of
(∗
s-ab
)]
the
projection
H
i
RTF
H
i
RTF-aug
may
be
identified
with
the
projection
H
i
RTF
(H
i
)
RTF
[which
is
an
isomorphism
whenever
Θ
i
=
Δ
i
].
Thus,
the
condition
(∗
s-ab
)
G
i
may
be
thought
of
as
the
condition
that
the
morphism
H
1
RTF
→
H
2
RTF
be
compatible
with
the
projection
morphisms
H
i
RTF
(H
i
)
RTF
G
i
.
From
this
point
of
view,
it
follows
immediately
that
the
semi-absoluteness
of
φ
implies
(∗
s-ab
),
and
that
(∗
s-ab
)
implies
[in
light
of
the
existence
of
H
1
,
H
2
—
cf.
Theorem
2.11,
(ii)]
the
pre-semi-
absoluteness
of
φ.
Assertion
(iii)
follows
from
Proposition
2.5,
(iv),
(v).
Remark
2.12.1.
The
criterion
of
Corollary
2.12,
(ii),
may
be
thought
of
as
a
“group-theoretic
Hom-version”,
in
the
case
of
hyperbolic
orbicurves,
of
the
numerical
criterion
“ζ(H)/ζ(Π)
=
[Π
:
H]”
of
Theorem
2.6,
(v).
Alternatively
[cf.
the
point
of
view
of
Remark
2.11.1],
this
criterion
of
Corollary
2.12,
(ii),
may
be
thought
of
as
a
[necessarily
—
cf.
Example
2.13
below!]
somewhat
more
complicated
version
for
MLF’s
of
the
latter
portion
of
Corollary
2.8,
(i),
in
the
case
of
FF’s
or
NF’s.
Example
2.13.
A
Non-pre-semi-absolute
Absolute
Homomorphism.
(i)
In
the
situation
of
Theorem
2.11,
suppose
that
Σ
=
Primes.
Fix
a
natural
number
N
[which
one
wants
to
think
of
as
being
“large”].
By
replacing
Π
by
an
open
subgroup
of
Π,
we
may
assume
that
Π
satisfies
the
hypotheses
of
the
subgroup
“H”
of
Theorem
2.11,
(iii),
and
that
the
dual
graph
of
the
special
fiber
of
X
is
not
a
tree
[cf.
the
discussion
preceding
[Mzk6],
Lemma
2.4].
Thus,
we
have
a
“combinatorial”
quotient
Π
Δ
com
,
where
Δ
com
is
a
nonabelian
profinite
free
group.
In
particular,
there
exists
an
open
subgroup
of
Δ
com
which
is
a
profinite
free
group
on
>
N
generators.
Thus,
by
replacing
Π
by
an
open
subgroup
of
Π
arising
from
an
open
subgroup
of
Δ
com
,
we
may
assume
from
the
start
that
Δ
com
is
a
profinite
free
group
on
>
N
generators.
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
I
(ii)
Now
let
33
1
→
Δ
∗
→
Π
∗
→
G
∗
→
1
be
an
extension
of
AFG-type
that
admits
a
construction
data
field
which
is
an
MLF.
Thus,
Π
∗
is
topologically
finitely
generated
[cf.
Theorem
2.6,
(ii)],
so
it
follows
that
there
exists
a
Π
as
in
(i),
together
with
a
surjection
of
profinite
groups
ψ
:
Π
Π
∗
that
factors
through
the
quotient
Π
Δ
com
.
In
particular,
ψ
is
an
absolute
homomorphism
which
is
not
pre-semi-absolute
[hence,
a
fortiori,
not
semi-absolute].
In
light
of
the
appearance
of
the
“combinatorial
quotient”
in
Theorem
2.11,
(iii),
we
pause
to
recall
the
following
result
[cf.
[Mzk6],
Lemma
2.3,
in
the
profinite
case].
Theorem
2.14.
(Graph-theoreticity
for
Hyperbolic
Curves)
For
i
=
1,
2,
let
1
→
Δ
i
→
Π
i
→
G
i
→
1,
k
i
,
X
i
,
Σ
i
,
φ
:
Π
1
→
Π
2
be
as
in
Proposition
2.5.
Suppose
further
that
k
i
is
an
MLF
of
residue
characteristic
p
i
;
that
Σ
i
contains
a
prime
=
p
i
;
that
φ
is
an
isomorphism;
and
that
X
i
is
a
hyperbolic
curve
with
stable
reduction
over
the
ring
of
integers
O
k
i
of
k
i
.
Write
Γ
i
for
the
dual
semi-
graph
with
compact
structure
[i.e.,
the
dual
graph,
together
with
additional
open
edges
corresponding
to
the
cusps
—
cf.
[Mzk6],
Appendix]
of
the
geometric
special
fiber
of
the
stable
model
of
X
i
over
O
k
i
.
Then:
∼
∼
(i)
We
have
p
1
=
p
2
,
Σ
1
=
Σ
2
;
φ
induces
isomorphisms
Δ
1
→
Δ
2
,
G
1
→
G
2
;
∼
φ
induces
an
isomorphism
of
semi-graphs
φ
Γ
:
Γ
1
→
Γ
2
which
is
functorial
in
φ.
In
particular,
the
natural
Galois
action
of
G
1
on
Γ
1
is
compatible,
relative
to
φ
Γ
,
with
the
natural
Galois
action
of
G
2
on
Γ
2
.
(ii)
For
i
=
1,
2,
suppose
that
the
action
of
G
i
on
Γ
i
is
trivial.
Write
Π
i
for
the
pro-Σ
i
“combinatorial”
quotient
determined
by
the
finite
Galois
coverings
of
the
semi-graph
Γ
i
of
degree
a
product
of
primes
∈
Σ
i
.
Then
φ
is
compatible
with
the
quotients
Π
i
Δ
com
.
i
Δ
com
i
Proof.
First,
we
consider
assertion
(i).
By
Corollary
2.8,
(ii),
p
1
=
p
2
,
and
φ
∼
∼
induces
isomorphisms
Δ
1
→
Δ
2
,
G
1
→
G
2
.
Since
[by
the
well-known
structure
of
geometric
fundamental
groups
of
hyperbolic
curves]
Σ
i
is
the
unique
minimal
Σ
⊆
Primes
such
that
Δ
i
is
almost
pro-Σ,
we
thus
conclude
that
Σ
1
=
Σ
2
.
Write
def
def
p
=
p
1
=
p
2
,
Σ
=
Σ
1
=
Σ
2
;
let
l
∈
Σ
be
such
that
l
=
p.
Then
it
follows
immediately
from
the
“Riemann
hypothesis
for
abelian
varieties
over
finite
fields”
[cf.,
e.g.,
[Mumf],
p.
206]
that
the
action
of
G
i
on
the
maximal
pro-l
quotient
Δ
i
(l)
Δ
i
is
—
in
the
terminology
of
[Mzk12]
—
“l-graphically
full”.
Thus,
by
[Mzk12],
(l)
∼
(l)
Corollary
2.7,
(ii),
the
isomorphism
Δ
1
→
Δ
2
is
—
again
in
the
terminology
of
[Mzk12]
—
“graphic”,
hence
induces
a
functorial
isomorphism
of
semi-graphs
∼
Γ
1
→
Γ
2
,
as
desired.
Next,
we
consider
assertion
(ii).
First,
we
observe
that,
by
assertion
(i),
the
condition
that
the
action
of
G
i
on
Γ
i
be
trivial
is
compatible
with
φ.
Also,
let
us
34
SHINICHI
MOCHIZUKI
observe
that
if
H
i
⊆
Π
i
is
an
open
subgroup
corresponding
to
a
finite
étale
covering
Z
i
→
X
i
of
X
i
,
then
the
condition
that
Z
i
have
stable
reduction
is
compatible
with
φ
[cf.
[Mzk6],
the
proof
of
Lemma
2.1;
our
assumption
that
there
exists
an
l
∈
Σ
i
such
that
l
=
p
i
].
Next,
I
claim
that:
if
and
only
A
finite
étale
Galois
covering
Z
i
→
X
i
of
X
i
arises
from
Δ
com
i
if
Z
i
has
stable
reduction,
and
the
action
of
Gal(Z
i
/X
i
)
on
the
dual
semi-
graph
with
compact
structure
of
the
geometric
special
fiber
of
the
stable
model
of
Z
i
is
free.
Indeed,
the
necessity
of
this
criterion
is
clear.
To
verify
the
sufficiency
of
this
criterion,
observe
that,
by
considering
the
non-free
actions
of
inertia
subgroups
of
the
Galois
covering
Z
i
→
X
i
,
it
follows
immediately
that
this
criterion
implies
that
all
of
the
inertia
groups
arising
from
irreducible
components
and
cusps
of
the
geometric
special
fiber
of
a
stable
model
of
X
i
are
trivial,
hence
[cf.,
e.g.,
[SGA2],
X,
3.4,
(i);
[Tama2],
Lemma
2.1,
(iii)]
that
the
covering
Z
i
→
X
i
extends
to
an
admissible
covering
of
the
respective
stable
models.
On
the
other
hand,
once
one
knows
that
the
covering
Z
i
→
X
i
admits
such
an
admissible
extension,
the
sufficiency
of
this
criterion
is
immediate.
This
completes
the
proof
of
the
claim.
Now
assertion
(ii)
follows
immediately,
by
applying
the
functorial
isomorphisms
of
semi-graphs
of
assertion
(i).
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
I
35
Section
3:
Absolute
Open
Homomorphisms
of
Local
Galois
Groups
In
the
present
§,
we
give
various
generalizations
of
the
main
result
of
[Mzk1]
concerning
isomorphisms
between
Galois
groups
of
MLF’s.
One
aspect
of
these
generalizations
is
the
substitution
of
the
condition
given
in
[Mzk1]
for
such
an
isomorphism
to
arise
geometrically
—
a
condition
that
involves
the
higher
ram-
ification
filtration
—
by
various
other
conditions
[cf.
Theorem
3.5].
Certain
of
these
conditions
were
motivated
by
a
recent
result
of
A.
Tamagawa
[cf.
Remark
3.8.1]
concerning
Lubin-Tate
groups
and
abelian
varieties
with
complex
multiplica-
tion;
other
conditions
[cf.
Corollary
3.7]
were
motivated
by
a
certain
application
of
the
theory
of
the
present
§3
to
be
discussed
in
[Mzk15].
Another
aspect
of
these
generalizations
is
that
certain
of
the
conditions
studied
below
allow
one
to
prove
a
“Hom-version”
[i.e.,
involving
open
homomorphisms,
as
opposed
to
just
isomor-
phisms
—
cf.
Theorem
3.5]
of
the
main
result
of
[Mzk1].
Finally,
this
Hom-version
of
the
main
result
of
[Mzk1]
implies
certain
semi-absolute
Hom-versions
[cf.
Corol-
lary
3.8,
3.9
below]
of
the
absolute
Isom-version
of
the
Grothendieck
Conjecture
given
in
[Mzk13],
§2,
and
the
relative
Hom-version
of
the
Grothendieck
Conjecture
for
function
fields
given
in
[Mzk3],
Theorem
B.
def
Let
k
be
an
MLF
of
residue
characteristic
p;
k
an
algebraic
closure
of
k;
G
k
=
k
the
p-adic
completion
of
k;
E
an
MLF
of
residue
characteristic
p
all
of
Gal(k/k);
whose
Q
p
-conjugates
are
contained
in
k.
Write
I
k
⊆
G
k
(respectively,
I
k
wild
⊆
I
k
)
for
def
=
G
k
/I
k
wild
;
the
inertia
subgroup
(respectively,
wild
inertia
subgroup)
of
G
k
;
G
tame
k
def
G
unr
=
G
k
/I
k
(
∼
=
Z).
k
Definition
3.1.
(i)
Let
A
be
an
abelian
topological
group;
ρ,
ρ
:
G
k
→
A
characters
[i.e.,
continuous
homomorphisms].
Then
we
shall
write
ρ
≡
ρ
and
say
that
ρ,
ρ
are
inertially
equivalent
if,
for
some
open
subgroup
H
⊆
I
k
,
the
restricted
characters
ρ|
H
,
ρ
|
H
coincide
[cf.
[Serre3],
III,
§A.5].
(ii)
Write
Emb(E,
k)
for
the
set
of
field
embeddings
σ
:
E
→
k.
Let
σ
∈
Emb(E,
k).
Then
if
π
is
a
uniformizer
of
k,
then
we
shall
denote
by
χ
σ,π
:
G
k
→
E
×
the
composite
homomorphism
∼
∼
O
×
→
O
×
⊆
E
×
→
(k
×
)
∧
→
O
k
×
×
Z
G
k
G
ab
k
E
k
∼
—
where
the
“∧”
denotes
the
profinite
completion;
the
first
“
→
”
is
the
isomor-
∼
phism
arising
from
local
class
field
theory
[cf.,
e.g.,
[Serre2]];
the
second
“
→
”
is
the
splitting
determined
by
π;
the
second
“”
is
the
projection
to
the
factor
O
k
×
,
composed
with
the
inverse
automorphism
on
O
k
×
[cf.
Remark
3.1.1
below];
×
is
the
norm
map
associated
to
the
field
embed-
the
homomorphism
O
k
×
→
O
E
ding
σ.
Since
[as
is
well-known,
from
local
class
field
theory]
I
k
⊆
G
k
surjects
to
it
follows
immediately
that
the
inertial
equivalence
class
of
O
k
×
×
{1}
⊆
O
k
×
×
Z,
χ
σ,π
is
independent
of
the
choice
of
π.
Thus,
we
shall
often
write
χ
σ
to
denote
χ
σ,π
for
some
unspecified
choice
of
π.
36
SHINICHI
MOCHIZUKI
(iii)
Let
ρ
:
G
k
→
E
×
be
a
character.
Then
we
shall
say
that
ρ
is
of
qLT-
type
[i.e.,
“quasi-Lubin-Tate”
type]
if
there
exists
an
open
subgroup
H
⊆
G
k
,
corresponding
to
a
field
extension
k
H
of
k,
and
a
field
embedding
σ
:
E
→
k
H
such
that
ρ|
H
≡
χ
σ
;
in
this
situation,
we
shall
refer
to
[E
:
Q
p
]
as
the
dimension
of
ρ.
We
shall
say
that
ρ
is
of
01-type
if
it
is
Hodge-Tate,
and,
moreover,
every
weight
appearing
in
its
Hodge-Tate
decomposition
∈
{0,
1}.
Write
χ
cyclo
:
G
k
→
Q
×
p
k
for
the
cyclotomic
character
associated
to
G
k
.
We
shall
say
that
ρ
is
of
ICD-type
[i.e.,
“inertially
cyclotomic
determinant”
type]
if
its
determinant
det(ρ)
:
G
k
→
Q
×
p
[i.e.,
the
composite
of
ρ
with
the
norm
map
E
×
→
Q
×
]
is
inertially
equivalent
to
p
.
χ
cyclo
k
(iv)
For
i
=
1,
2,
let
k
i
be
an
MLF
of
residue
characteristic
p
i
;
k
i
an
algebraic
k
the
p
-adic
completion
of
k
.
We
shall
use
similar
notation
for
the
closure
of
k
;
i
i
i
i
def
various
subquotients
of
the
absolute
Galois
group
G
k
i
=
Gal(k
i
/k
i
)
of
k
i
to
the
notation
already
introduced
for
G
k
.
Let
φ
:
G
k
1
→
G
k
2
be
an
open
homomorphism.
Then
we
shall
say
that
φ
is
of
qLT-type
(respectively,
of
01-qLT-type)
if
p
1
=
p
2
,
and,
moreover,
for
every
pair
of
open
subgroups
H
1
⊆
G
k
1
,
H
2
⊆
G
k
2
such
that
φ(H
1
)
⊆
H
2
,
and
every
character
ρ
:
H
2
→
F
×
of
qLT-type
[where
F
is
an
MLF
of
residue
characteristic
p
1
=
p
2
all
of
whose
conjugates
are
contained
in
the
fields
determined
by
H
1
,
H
2
],
the
restricted
character
ρ|
H
1
:
H
1
→
F
×
[obtained
by
restricting
via
φ]
is
of
qLT-type
(respectively,
of
01-type).
We
shall
say
that
φ
is
of
HT-type
[i.e.,
“Hodge-Tate”
type]
if
p
1
=
p
2
,
and,
moreover,
the
topological
G
k
1
-module
[but
not
necessarily
the
topological
field!]
obtained
by
k
is
isomorphic
[as
a
topological
composing
φ
with
the
natural
action
of
G
on
k
2
2
k
1
.
We
shall
say
that
φ
is
of
CHT-type
[i.e.,
“cyclotomic
Hodge-
G
k
1
-module]
to
Tate”
type]
if
φ
is
of
HT-type,
and,
moreover,
the
cyclotomic
characters
of
G
k
1
,
=
χ
cyclo
◦
φ.
We
shall
say
that
φ
is
geometric
if
it
arises
from
an
G
k
2
satisfy
χ
cyclo
k
1
k
2
∼
isomorphism
of
fields
k
2
→
k
1
that
maps
k
2
into
k
1
[which
implies,
by
considering
∼
the
divisibility
of
the
k
i
×
,
that
p
1
=
p
2
,
and
that
the
isomorphism
k
2
→
k
1
is
compatible
with
the
respective
topologies].
(v)
Let
1
→
Δ
→
Π
→
G
k
→
1
be
an
extension
of
AFG-type.
Then
we
shall
say
that
this
extension
Π
G
k
[or,
when
there
is
no
danger
of
confusion,
that
Π]
is
of
A-qLT-type
[i.e.,
“Albanese-quasi-Lubin-Tate”
type]
if
for
every
open
subgroup
H
⊆
G
k
,
and
every
character
ρ
:
H
→
F
×
of
qLT-type
[where
F
is
an
MLF
of
residue
characteristic
p
all
of
whose
conjugates
are
contained
in
the
field
determined
by
H],
there
exists
an
open
subgroup
J
⊆
Π
×
G
k
(I
k
H)
[so
one
has
an
outer
def
action
of
the
image
J
G
of
J
in
G
k
on
J
Δ
=
J
Δ]
such
that
the
J
G
-module
V
ρ
obtained
by
letting
J
G
act
on
F
via
ρ|
J
G
is
isomorphic
to
some
subquotient
S
of
ab
the
J
G
-module
J
Δ
⊗
Q
p
.
Remark
3.1.1.
As
is
well-known,
the
ρ
that
arises
from
a
Lubin-Tate
group
is
of
qLT-type
—
cf.,
e.g.,
[Serre3],
III,
§A.4,
Proposition
4.
This
is
the
reason
for
the
terminology
“quasi-Lubin-Tate”.
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
I
37
We
begin
by
reviewing
some
well-known
facts.
Proposition
3.2.
(Characterization
of
Hodge-Tate
Characters)
Let
ρ
:
×
G
k
→
E
be
a
character;
write
V
ρ
for
the
G
k
-module
obtained
by
letting
G
k
act
on
E
via
ρ.
Then
ρ
is
Hodge-Tate
if
and
only
if
χ
n
σ
σ
ρ
≡
σ∈Emb(E,k)
for
some
n
σ
∈
Z.
Moreover,
in
this
case,
we
have
an
isomorphism
of
k[G
k
]-modules:
k
∼
V
ρ
⊗
Q
p
=
k(n
σ
)
σ∈Emb(E,k)
[where
the
“(−)”
denotes
a
Tate
twist].
Proof.
Indeed,
this
criterion
for
the
character
ρ
to
be
Hodge-Tate
is
precisely
the
content
of
[Serre3],
III,
§A.5,
Corollary.
The
Hodge-Tate
decomposition
of
V
ρ
then
follows
immediately
the
Hodge-Tate
decomposition
of
“V
ρ
”
in
the
case
where
one
takes
“ρ”
to
be
χ
σ
[cf.
[Serre3],
III,
§A.5,
proof
of
Lemma
2].
Proposition
3.3.
(Characterization
of
Quasi-Lubin-Tate
Characters)
Let
ρ,
V
ρ
be
as
in
Proposition
3.2.
Then
the
following
conditions
on
ρ
are
equiv-
alent:
(i)
ρ
is
of
qLT-type.
(ii)
We
have
an
isomorphism
of
k[G
k
]-modules:
V
ρ
⊗
Q
p
k
∼
=
k(1)
⊕
k
⊕
.
.
.
⊕
k.
(iii)
ρ
is
of
ICD-type
and
Hodge-Tate;
the
resulting
n
σ
’s
of
Proposition
3.2
are
∈
{0,
1}.
(iv)
ρ
is
of
ICD-type
and
of
01-type.
Proof.
The
fact
that
(i)
implies
(ii)
follows
immediately
from
the
description
of
the
Hodge-Tate
decomposition
of
“V
ρ
”
in
the
case
where
one
takes
“ρ”
to
be
χ
σ
[cf.
[Serre3],
III,
§A.5,
proof
of
Lemma
2].
Next,
let
us
assume
that
(ii),
(iii),
or
(iv)
holds.
In
either
of
these
cases,
it
follows
that
ρ,
hence
also
the
determinant
det(ρ)
:
G
k
→
Q
×
p
of
ρ,
is
Hodge-Tate.
Then
by
applying
Proposition
3.2
to
ρ,
we
obtain
that
the
associated
n
σ
’s
are
∈
{0,
1};
by
applying
Proposition
3.2
to
det(ρ)
[in
which
case
one
takes
“E”
to
be
Q
p
],
we
obtain
that
det(ρ)
is
inertially
equivalent
to
the
(
σ
n
σ
)-th
power
of
χ
cyclo
.
But
this
allows
one
to
conclude
[either
from
k
the
explicit
Hodge-Tate
decomposition
of
(ii),
or
from
the
assumption
that
ρ
is
of
ICD-type
in
(iii),
(iv)]
that
σ
n
σ
=
1,
hence
that
there
exists
precisely
one
σ
∈
Emb(E,
k)
such
that
n
σ
=
1,
n
σ
=
0
for
σ
=
σ.
Thus,
[sorting
through
the
definitions]
we
conclude
that
(i),
(ii),
(iii),
and
(iv)
hold.
This
completes
the
proof
of
Proposition
3.3.
38
SHINICHI
MOCHIZUKI
Proposition
3.4.
(Preservation
of
Tame
Quotients)
In
the
notation
of
Definition
3.1,
(iv),
let
φ
:
G
k
1
→
G
k
2
be
an
open
homomorphism.
Then
p
1
=
p
2
,
and
there
exists
a
commutative
diagram
G
k
1
⏐
⏐
G
tame
k
1
φ
−→
φ
tame
−→
G
k
2
⏐
⏐
G
tame
k
2
—
where
the
vertical
arrows
are
the
natural
surjections;
φ
tame
is
an
injective
homomorphism.
Proof.
We
may
assume
without
loss
of
generality
that
φ
is
surjective.
Next,
let
def
H
2
⊆
G
k
2
be
an
open
subgroup,
H
1
=
φ
−1
(H
2
)
⊆
G
k
1
.
Then
if
p
1
=
p
2
,
then
[since
we
have
a
surjection
H
2
H
1
]
1
=
δ
l
1
(H
2
)
≥
δ
l
1
(H
1
)
≥
2
for
l
=
p
1
[cf.
def
Theorem
2.6,
(ii)];
thus,
we
conclude
that
p
1
=
p
2
.
Write
p
=
p
1
=
p
2
.
Since
∼
[for
some
faithful
action
of
Z
on
Z
(
=
p)
(1)
—
cf.,
e.g.,
[NSW],
(
=
p)
(1)
Z
G
tame
=
Z
k
2
Theorem
7.5.2],
it
follows
immediately
that
every
closed
normal
pro-p
subgroup
of
G
tame
is
trivial.
Thus,
the
image
of
φ(I
k
wild
)
in
G
tame
is
trivial,
so
we
conclude
that
k
2
k
2
1
tame
φ
induces
a
surjection
φ
tame
:
G
tame
G
.
Since,
for
i
=
1,
2,
the
quotient
k
1
k
2
tame
∼
G
tame
G
unr
(G
tame
)
ab-t
,
it
k
i
k
i
=
Z
may
be
characterized
as
the
quotient
G
k
i
k
i
tame
thus
follows
immediately
that
φ
induces
continuous
homomorphisms
∼
(
=
p)
(1)
∼
(
=
p)
(1)
→
I
k
2
/I
k
wild
Z
=
I
k
1
/I
k
wild
=
Z
1
2
unr
∼
∼
Z
=
G
unr
k
1
G
k
2
=
Z;
—
the
first
of
which
is
surjective,
hence
an
isomorphism
[since,
as
is
well-known,
every
surjective
endomorphism
of
a
topologically
finitely
generated
profinite
group
is
an
isomorphism].
But
this
implies
that
the
second
displayed
homomorphism
is
also
surjective,
hence
an
isomorphism.
This
completes
the
proof
of
Proposition
3.4.
(Criteria
for
Geometricity)
For
i
=
1,
2,
let
k
i
be
an
MLF
k
the
p
-adic
completion
of
residue
characteristic
p
;
k
an
algebraic
closure
of
k
;
Theorem
3.5.
i
i
i
i
i
of
k
i
.
We
shall
use
similar
notation
for
the
various
subquotients
of
the
absolute
def
Galois
group
G
k
i
=
Gal(k
i
/k
i
)
of
k
i
to
the
notation
introduced
at
the
beginning
of
the
present
§3
for
G
k
.
Let
φ
:
G
k
1
→
G
k
2
be
an
open
homomorphism.
Then:
(i)
The
following
conditions
on
φ
are
equivalent:
(a)
φ
is
of
CHT-type;
(b)
φ
is
of
01-qLT-type;
(c)
φ
is
of
qLT-type;
(d)
φ
is
geometric.
(ii)
Suppose
that
φ
is
an
isomorphism.
Then
φ
is
geometric
if
and
only
if
it
is
of
HT-type.
(iii)
For
i
=
1,
2,
let
1
→
Δ
i
→
Π
i
→
G
k
i
→
1
be
an
extension
of
AFG-type;
ψ
:
Π
1
→
Π
2
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
I
39
a
semi-absolute
[or,
equivalently,
pre-semi-absolute
—
cf.
Proposition
2.5,
(iii)]
homomorphism
that
lifts
φ.
Suppose
that
Π
2
is
of
A-qLT-type.
Then
φ
is
geometric.
Proof.
First,
we
observe
that
by
Proposition
3.4,
it
follows
that
p
1
=
p
2
;
write
def
p
=
p
1
=
p
2
.
Also,
we
may
always
assume
without
loss
of
generality
that
φ
is
surjective.
Thus,
by
Proposition
3.4,
it
follows
that
φ(I
k
1
)
=
I
k
2
.
In
the
following,
we
will
use
a
superscript
“G
k
i
”
[where
i
=
1,
2]
to
denote
the
submodule
of
G
k
i
-
invariants
of
a
G
k
i
-module.
Next,
we
consider
assertion
(i).
First,
we
observe
that
it
is
immediate
that
condition
(d)
implies
condition
(a).
Next,
let
us
suppose
that
condition
(a)
holds.
G
ki
Since
k
=
k
is
finite-dimensional
over
Q
,
it
follows
that,
for
i
=
1,
2,
any
G
-
i
i
p
k
i
module
M
which
is
finite-dimensional
over
Q
p
is
Hodge-Tate
with
weights
∈
{0,
1}
if
and
only
if
G
k
i
dim
Q
p
((M
⊗
k
i
)
G
ki
)
+
dim
Q
p
((M
(−1)
⊗
k
i
)
G
ki
)
=
dim
Q
p
(M
)
·
dim
Q
p
(
k
i
)
[where
the
tensor
products
are
over
Q
p
].
Now
suppose
that
M
is
a
G
k
2
-module
that
arises
as
a
“V
ρ
”
for
some
character
ρ
:
G
k
2
→
E
×
of
qLT-type
[so
M
is
Hodge-Tate
with
weights
∈
{0,
1}
—
cf.
Proposition
3.3,
(i)
=⇒
(iv)];
write
M
φ
for
the
G
k
1
-module
M
φ
obtained
by
composing
the
G
k
2
-action
on
M
with
φ.
Thus,
it
follows
immediately
from
our
assumption
that
φ
is
of
CHT-type
that
the
above
condition
concerning
Q
p
-dimensions
for
M
implies
the
above
condition
concerning
Q
p
-dimensions
for
M
φ
.
Applying
this
argument
to
corresponding
open
subgroups
of
G
k
1
,
G
k
2
thus
shows
that
φ
is
of
01-qLT-type,
i.e.,
that
condition
(b)
holds.
≡
χ
cyclo
◦
Next,
let
us
assume
that
condition
(b)
holds.
First,
I
claim
that
χ
cyclo
k
1
k
2
cyclo
×
φ.
Indeed,
by
condition
(b),
it
follows
that
the
character
χ
k
2
◦
φ
:
G
k
1
→
Q
p
is
n
of
01-type.
Thus,
by
Proposition
3.2,
we
conclude
that
χ
cyclo
◦
φ
≡
(χ
cyclo
k
2
k
1
)
,
for
some
n
∈
{0,
1}.
On
the
other
hand,
the
restriction
of
χ
cyclo
to
I
k
2
clearly
has
open
k
2
image;
since
φ(I
k
1
)
=
I
k
2
,
it
thus
follows
that
the
restriction
of
χ
cyclo
◦
φ
to
I
k
1
has
k
2
open
image.
This
rules
out
the
possibility
that
n
=
0,
hence
completes
the
proof
of
the
claim.
Now,
by
applying
this
claim,
together
with
Proposition
3.3,
(i)
⇐⇒
(iv),
we
conclude
that
φ
is
of
qLT-type,
i.e.,
that
condition
(c)
holds.
Next,
let
us
assume
that
condition
(c)
holds.
First,
I
claim
that
this
already
implies
that
φ
is
injective
[i.e.,
an
isomorphism].
Indeed,
let
γ
∈
Ker(φ)
⊆
G
k
1
be
such
that
γ
=
1.
Then
there
exists
an
open
subgroup
J
1
⊆
G
k
1
⊆
G
Q
p
satisfying
the
following
conditions:
(1)
γ
∈
/
J
1
;
(2)
J
1
is
characteristic
as
a
subgroup
G
Q
p
;
(3)
the
extension
E
of
Q
p
determined
by
J
1
contains
all
Q
p
-conjugates
of
k
2
.
Fix
an
embedding
σ
0
:
k
2
→
E;
write
H
2
⊆
G
k
2
for
the
corresponding
open
subgroup.
Let
H
1
⊆
J
1
⊆
G
k
1
be
an
open
subgroup
which
is
normal
in
G
k
1
such
that
φ(H
1
)
⊆
H
2
;
for
i
=
1,
2,
write
k
H
i
for
the
extension
of
k
i
determined
by
H
i
.
Thus,
the
embedding
σ
2
:
E
→
k
H
2
given
by
the
identity
E
=
k
H
2
(respectively,
σ
1
:
E
→
k
H
1
determined
by
the
inclusion
H
1
⊆
J
1
)
determines
a
character
ρ
2
:
H
2
→
E
×
(respectively,
ρ
1
:
H
1
→
E
×
)
of
qLT-type
[i.e.,
the
character
“χ
σ
2
”
(respectively,
“χ
σ
1
”)].
Moreover,
by
condition
(c),
the
character
ρ
2
◦
(φ|
H
1
)
:
H
1
→
E
×
is
of
qLT-
type,
hence
is
inertially
equivalent
to
τ
◦
ρ
1
:
H
1
→
E
×
for
some
τ
∈
Gal(E/Q
p
).
In
40
SHINICHI
MOCHIZUKI
particular,
by
replacing
σ
2
by
σ
2
◦
τ
−1
,
we
may
assume
that
τ
is
the
identity,
hence
that
ρ
2
◦
(φ|
H
1
)
≡
ρ
1
.
On
the
other
hand,
since
γ
∈
/
J
1
,
hence
acts
nontrivially
on
the
subfield
E
⊆
k
H
1
[relative
to
the
embedding
σ
1
],
it
follows
that
ρ
1
◦
κ
γ
≡
δ
◦
ρ
1
,
where
we
write
κ
γ
for
the
automorphism
of
H
1
given
by
conjugating
by
γ,
and
δ
∈
Gal(E/Q
p
)
is
not
equal
to
the
identity.
But
since
φ(γ)
=
1
∈
G
k
2
,
we
thus
conclude
that
δ
◦
ρ
1
≡
ρ
1
◦
κ
γ
≡
ρ
2
◦
(φ|
H
1
)
◦
κ
γ
≡
ρ
2
◦
(φ|
H
1
)
≡
ρ
1
,
which
[since
ρ
1
has
open
image]
contradicts
the
fact
that
δ
∈
Gal(E/Q
p
)
is
not
equal
to
the
identity.
This
completes
the
proof
of
the
claim.
Thus,
we
may
assume
that
φ
is
an
isomorphism
of
qLT-type,
i.e.,
we
are,
in
effect,
in
the
situation
of
[Mzk1],
§4.
In
particular,
the
fact
that
φ
is
geometric,
i.e.,
that
condition
(d)
holds,
follows
immediately
via
the
argument
of
[Mzk1],
§4.
This
completes
the
proof
of
assertion
(i).
Next,
we
consider
assertion
(ii).
Since
φ
is
an
isomorphism,
it
follows
[cf.
[Mzk1],
Proposition
1.1;
[Mzk6],
Proposition
1.2.1,
(vi)]
that
χ
cyclo
=
χ
cyclo
◦
φ.
In
k
1
k
2
particular,
φ
is
of
HT-type
if
and
only
if
φ
is
of
CHT-type.
Thus,
assertion
(ii)
follows
from
the
equivalence
of
(a),
(d)
in
assertion
(i).
Finally,
we
consider
assertion
(iii).
First,
let
us
recall
that
by
a
well-known
result
of
Tate
[cf.
[Tate],
§4,
Corollary
2],
if
J
⊆
Π
1
is
an
open
subgroup
with
image
def
ab
⊗
Q
p
is
always
J
G
⊆
G
k
1
and
intersection
J
Δ
=
J
Δ
1
,
then
the
J
G
-module
J
Δ
Hodge-Tate
with
weights
∈
{0,
1}.
Thus,
the
condition
that
Π
2
is
of
A-qLT-type
implies
that
φ
is
of
01-qLT-type,
hence,
by
assertion
(i),
geometric.
This
completes
the
proof
of
assertion
(iii).
Definition
3.6.
(i)
If
H
⊆
G
k
is
an
open
subgroup
corresponding
to
an
extension
field
k
H
of
k,
then
by
local
class
field
theory
[cf.,
e.g.,
[Serre2]],
we
have
a
natural
isomorphism
∼
O
k
×
H
→
Tor(H)
—
where
we
write
Tor(H)
[i.e.,
the
“toral
portion
of
H”]
for
the
image
of
I
k
H
in
H
ab
.
Thus,
by
applying
the
p-adic
logarithm
O
k
×
H
→
k
H
,
we
obtain
a
natural
∼
isomorphism
λ
H
:
Tor(H)
⊗
Q
p
→
k
H
.
(ii)
We
shall
refer
to
a
collection
{N
H
}
H
,
where
H
ranges
over
a
collection
of
open
subgroups
of
G
k
that
form
a
basis
of
the
topology
of
G
k
,
as
a
uniformly
toral
neighborhood
of
G
k
if
there
exist
nonnegative
integers
a,
b
[which
are
independent
of
H!]
such
that
[in
the
notation
of
(i)]
N
H
⊆
Tor(H)
⊗
Q
p
is
an
open
subgroup
such
that
p
a
·
O
k
H
⊆
λ
H
(N
H
)
⊆
p
−b
·
O
k
H
⊆
k
H
.
∼
(iii)
Let
φ
:
G
k
1
→
G
k
2
be
an
isomorphism
of
profinite
groups.
Then
we
shall
say
that
φ
is
uniformly
toral
if
G
k
1
admits
a
uniformly
toral
neighborhood
{N
H
}
H
def
such
that
{N
φ(H)
=
φ(N
H
)}
φ(H)
forms
a
uniformly
toral
neighborhood
of
G
k
2
.
We
shall
say
that
φ
is
RF-preserving
[i.e.,
“ramification
filtration
preserving”]
if
φ
is
compatible
with
the
filtrations
on
G
k
1
,
G
k
2
given
by
the
[positively
indexed]
higher
ramification
groups
in
the
upper
numbering
[cf.,
[Mzk1],
Theorem].
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
I
41
Corollary
3.7.
(Uniform
Torality
and
Geometricity)
In
the
situation
of
Theorem
3.5,
suppose
further
that
φ
is
an
isomorphism.
Then
the
following
conditions
on
φ
are
equivalent:
(a)
φ
is
RF-preserving;
(b)
φ
is
uniformly
toral;
(c)
φ
is
geometric.
Proof.
First,
we
observe
that
by
Proposition
3.4,
it
follows
that
p
1
=
p
2
;
write
def
p
=
p
1
=
p
2
.
Also,
we
observe
that
it
is
immediate
that
condition
(c)
implies
condition
(a).
Next,
we
recall
that
the
fact
that
condition
(a)
implies
condition
(b)
is
precisely
the
content
of
the
discussion
preceding
[Mzk1],
Proposition
2.2.
That
is
to
say,
for
i
=
1,
2,
the
images
of
appropriate
higher
ramification
groups
in
Tor(H)
⊗
Q
p
[for
open
subgroups
H
⊆
G
k
i
]
multiplied
by
appropriate
integral
powers
of
p
yield
a
uniformly
toral
neighborhood
of
G
k
i
that
is
compatible
with
φ
whenever
φ
is
RF-preserving.
i
Next,
let
us
assume
that
condition
(b)
holds.
For
i
=
1,
2,
let
{N
H
}
H
be
a
uniformly
toral
neighborhood
of
G
k
i
.
Again,
we
take
the
point
of
view
of
the
discussion
preceding
[Mzk1],
Proposition
2.2.
That
is
to
say,
we
think
of
k
i
as
the
inductive
limit
def
I
i
=
lim
−→
Tor(H)
⊗
Q
p
H
i
—
where
H
ranges
over
the
open
subgroups
⊆
G
k
i
involved
in
{N
H
}
H
;
the
mor-
phisms
in
the
inductive
system
are
those
induced
by
the
Verlagerung,
or
transfer,
i
map.
Write
N
i
⊆
I
i
for
the
subgroup
generated
by
the
N
H
⊆
Tor(H)
⊗
Q
p
.
Then
∼
relative
to
the
isomorphism
[of
abstract
modules!]
λ
i
:
I
i
→
k
i
determined
by
the
λ
H
’s,
we
have
p
a
·
O
k
i
⊆
λ
i
(N
i
)
⊆
p
−b
·
O
k
i
⊆
k
i
for
some
nonnegative
integers
a,
b
[cf.
Definition
3.6,
(ii)].
In
particular,
it
follows
that
the
topology
on
I
i
determined
by
the
submodules
p
c
·
N
i
,
where
c
≥
0
is
an
integer,
coincides,
relative
to
λ
i
,
with
the
p-adic
topology
on
k
i
[i.e.,
the
topology
determined
by
the
p
c
·
O
k
i
,
where
c
≥
0
is
an
integer].
Write
I
i
for
the
completion
of
I
i
relative
to
the
topology
determined
by
the
p
c
·
N
i
.
Thus,
λ
i
determines
an
∼
k
i
.
In
particular,
the
assumption
that
isomorphism
of
topological
G
k
i
-modules
I
i
→
φ
is
uniformly
toral
implies
that
φ
is
of
HT-type.
Thus,
by
Theorem
3.5,
(ii),
we
conclude
that
φ
is
geometric,
i.e.,
that
condition
(c)
holds.
This
completes
the
proof
of
Corollary
3.7.
Remark
3.7.1.
In
fact,
one
verifies
immediately
that
the
argument
applied
in
the
proof
of
Corollary
3.7
implies
that
the
equivalences
of
Corollary
3.7
[as
well
as
the
definitions
of
Definition
3.6]
continue
to
hold
when
φ
is
replaced
by
an
isomorphism
of
profinite
groups
between
the
maximal
pro-p
quotients
of
the
G
k
i
.
We
leave
the
routine
details
to
the
reader.
Corollary
3.8.
(Geometricity
of
Semi-absolute
Homomorphisms
for
k
,
p
,
G
[and
its
subquotients]
Hyperbolic
Orbicurves)
For
i
=
1,
2,
let
k
,
k
,
i
i
i
i
k
i
be
as
in
Theorem
3.5;
1
→
Δ
i
→
Π
i
→
G
k
i
→
1
an
extension
of
AFG-type;
(k
i
,
X
i
,
Σ
i
)
partial
construction
data
[consisting
of
the
construction
data
field,
42
SHINICHI
MOCHIZUKI
construction
data
base-stack,
and
construction
data
prime
set]
for
Π
i
G
k
i
;
α
i
:
π
1
(X
i
)
=
π
1
tame
(X
i
)
Π
i
a
scheme-theoretic
envelope
compatible
with
the
natural
projections
π
1
(X
i
)
G
k
i
,
Π
i
G
k
i
;
ψ
:
Π
1
→
Π
2
a
semi-absolute
[or,
equivalently,
pre-semi-absolute
—
cf.
Proposition
2.5,
(iii)]
homomorphism
that
lifts
a
homomorphism
φ
:
G
1
→
G
2
.
Suppose
further
that
X
2
is
a
hyperbolic
orbicurve,
that
p
2
∈
Σ
2
,
and
that
one
of
the
following
conditions
holds:
(a)
φ
is
of
CHT-type;
(b)
φ
is
of
01-qLT-type;
(c)
φ
is
of
qLT-type;
(d)
φ
is
an
isomorphism
of
HT-type;
(e)
φ
is
a
uniformly
toral
isomorphism;
(f
)
φ
is
an
RF-preserving
isomorphism;
(g)
Π
2
is
of
A-qLT-type.
(h)
φ
is
geometric;
Then
ψ
is
geometric,
i.e.,
arises
[relative
to
the
α
i
]
from
a
unique
dominant
morphism
of
schemes
X
1
→
X
2
lying
over
a
morphism
Spec(k
1
)
→
Spec(k
2
).
Proof.
Indeed,
by
Theorem
3.5,
(i),
(ii),
(iii);
Corollary
3.7,
it
follows
that
any
of
the
conditions
(a),
(b),
(c),
(d),
(e),
(f),
(g),
(h)
implies
condition
(h).
Thus,
since
X
2
is
a
hyperbolic
orbicurve,
and
p
2
∈
Σ
2
,
the
fact
that
ψ
is
geometric
follows
from
[Mzk3],
Theorem
A.
Remark
3.8.1.
One
important
motivation
for
the
theory
of
the
present
§3
is
the
following
result,
orally
communicated
to
the
author
by
A.
Tamagawa:
(∗
A-qLT
)
Let
X
be
a
hyperbolic
orbicurve
over
k
that
admits
a
finite
étale
covering
Y
→
X
by
a
hyperbolic
curve
Y
such
that
Y
admits
a
dominant
k-
morphism
Y
→
P
,
where
P
is
the
projective
line
minus
three
points
over
k
[i.e.,
a
tripod
—
cf.
§0].
Then
the
arithmetic
fundamental
group
π
1
(X)
G
k
of
X
is
of
A-qLT-type.
In
particular,
it
follows
that:
Corollary
3.8
may
be
applied
[in
the
sense
that
condition
(g)
is
satisfied]
whenever
X
2
satisfies
the
conditions
placed
on
the
hyperbolic
orbicurve
“X”
of
(∗
A-qLT
).
Indeed,
Tamagawa’s
original
motivation
for
considering
(∗
A-qLT
)
was
precisely
the
goal
of
applying
the
methods
of
[Mzk1]
to
obtain
an
“isomorphism
version”
of
Corollary
3.8,
(g).
Upon
learning
of
these
ideas
of
Tamagawa,
the
author
proceeded
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
I
43
to
re-examine
the
theory
of
[Mzk1].
This
led
the
author
to
the
discovery
of
the
various
generalizations
of
[Mzk1]
—
and,
in
particular,
the
Hom-version
of
Corollary
3.8,
(g)
—
given
in
the
present
§3.
Tamagawa
derives
(∗
A-qLT
)
from
the
following
result:
(∗
CM
)
Given
a
character
ρ
:
G
k
→
E
×
of
qLT-type,
there
exists
an
abelian
variety
with
complex
multiplication
A
over
some
finite
extension
k
A
of
k
such
that
ρ|
G
kA
is
inertially
equivalent
to
some
character
whose
associated
G
k
A
-module
appears
as
a
subquotient
of
the
G
k
A
-module
given
by
the
p-
adic
Tate
module
of
A.
Indeed,
to
derive
(∗
A-qLT
)
from
(∗
CM
),
one
reasons
as
follows:
Every
abelian
variety
with
complex
multiplication
A
is
defined
over
a
number
field,
hence
arises
as
a
quotient
of
a
Jacobian
of
a
smooth
proper
curve
Z
over
a
number
field.
Moreover,
by
considering
Belyi
maps,
it
follows
that
some
open
subscheme
U
Z
⊆
Z
arises
as
a
finite
étale
covering
of
the
projective
line
minus
three
points.
Thus,
any
Galois
module
that
appears
as
a
subquotient
of
the
p-adic
Tate
module
of
A
also
appears
as
a
subquotient
of
the
p-adic
Tate
module
of
the
Jacobian
of
some
finite
étale
covering
of
the
curve
P
of
(∗
A-qLT
),
hence,
a
fortiori,
as
a
subquotient
of
the
p-adic
Tate
module
of
the
Jacobian
of
some
finite
étale
covering
of
the
curves
Y
,
X
of
(∗
A-qLT
).
Thus,
we
conclude
that
π
1
(X)
is
of
A-qLT-type,
as
desired.
Corollary
3.9.
(Geometricity
of
Strictly
Semi-absolute
Homomor-
phisms
for
Function
Fields)
Assume
that
either
of
the
results
(∗
A-qLT
),
(∗
CM
)
of
Remark
3.8.1
holds.
For
i
=
1,
2,
let
k
i
be
an
MLF,
K
i
a
function
field
of
tran-
scendence
degree
≥
1
over
k
i
[so
k
i
is
algebraically
closed
in
K
i
],
K
i
an
algebraic
def
closure
of
K
i
,
k
i
the
algebraic
closure
of
k
i
determined
by
K
i
,
Π
i
=
Gal(K
i
/K
i
),
def
def
G
i
=
Gal(k
i
/k
i
),
Δ
i
=
Ker(Π
i
G
i
).
Then
every
open
homomorphism
ψ
:
Π
1
→
Π
2
that
induces
an
open
homomorphism
ψ
Δ
:
Δ
1
→
Δ
2
[hence
also
an
open
homomor-
phism
φ
:
G
1
→
G
2
]
is
geometric,
i.e.,
arises
from
a
unique
embedding
of
fields
K
2
→
K
1
that
induces
an
embedding
of
fields
k
2
→
k
1
of
finite
degree.
Proof.
Since
every
function
field
of
transcendence
degree
≥
1
over
k
2
contains
the
function
field
of
a
tripod
over
k
2
,
it
follows
from
(∗
A-qLT
),
hence
also
from
(∗
CM
)
[cf.
Remark
3.8.1],
that
there
exists
a
hyperbolic
curve
X
over
k
2
whose
function
def
field
is
contained
in
K
2
such
that
if
we
write
Π
2
Π
3
=
π
1
(X)
for
the
resulting
surjection,
then
Π
3
is
of
A-qLT-type.
Now
we
wish
to
apply
a
“birational
analogue”
of
Corollary
3.8,
(g),
to
the
composite
homomorphism
Π
1
→
Π
2
Π
3
[where
the
first
arrow
is
ψ].
To
verify
that
such
an
analogue
holds,
it
suffices
to
verify
that
φ
is
of
01-qLT-
def
def
type
[cf.
Theorem
3.5,
(i),
(b)
=⇒
(d)].
To
this
end,
set
k
3
=
k
2
,
G
3
=
G
2
,
def
Δ
3
=
Ker(Π
3
G
3
);
let
us
suppose,
for
i
=
1,
3,
that
H
i
⊆
Δ
i
,
J
i
⊆
G
i
are
characteristic
open
subgroups
such
that
ψ
Δ
(H
1
)
⊆
H
3
,
φ(J
1
)
⊆
J
3
.
Thus,
if
44
SHINICHI
MOCHIZUKI
we
write
p
for
the
common
residue
characteristic
of
k
1
,
k
3
[cf.
Proposition
3.4],
then
we
obtain
a
surjection
H
1
ab
⊗
Q
p
H
3
ab
⊗
Q
p
that
is
compatible
with
φ.
Moreover,
it
follows
immediately
from
Corollary
A.11
[cf.
also
Proposition
A.3,
(v)]
of
the
Appendix
that
the
J
1
-module
H
1
ab-t
⊗
Z
p
admits
a
quotient
J
1
-module
H
1
ab-t
⊗
Z
p
Q
1
such
that
Q
1
is
the
p-adic
Tate
module
of
some
abelian
variety
over
a
finite
extension
of
k
1
,
and,
moreover,
the
kernel
Ker(H
1
ab-t
⊗
Z
p
Q
1
)
is
topologically
generated
by
topologically
cyclic
subgroups
[i.e.,
“copies
of
Z
p
”]
on
which
some
open
subgroup
of
J
1
[which
may
depend
on
the
cyclic
subgroup]
acts
via
the
cyclotomic
character.
Next,
let
us
observe
that
if
V
3
is
any
J
3
-module
associated
to
a
character
of
qLT-type
of
dimension
≥
2,
then
V
3
does
not
contain
any
sub-J
3
-
modules
of
dimension
1
over
Q
p
.
From
this
observation,
it
follows
immediately
that
any
subquotient
[cf.
Definition
3.1,
(v)]
of
the
J
3
-module
H
3
ab
⊗Q
p
that
is
isomorphic
to
the
J
3
-module
associated
to
a
character
of
qLT-type
of
dimension
≥
2
determines
a
subquotient
[not
only
of
the
J
1
-module
H
1
ab-t
⊗
Q
p
,
but
also]
of
the
J
1
-module
Q
1
⊗
Q
p
.
Thus,
we
conclude
that
any
such
subquotient
of
the
J
1
-module
Q
1
⊗
Q
p
is
Hodge-Tate
with
weights
∈
{0,
1}.
Moreover,
by
considering
determinants
of
such
subquotients,
one
concludes
that
the
pull-back
of
the
cyclotomic
character
J
3
→
Z
×
p
is
a
character
J
1
→
Z
×
p
which
is
Hodge-Tate,
and
whose
unique
weight
w
is
≥
0.
If
w
≥
2,
then
the
fact
that
the
J
3
-module
determined
by
the
cyclotomic
character
of
J
3
occurs
as
a
subquotient
of
H
3
ab
⊗
Q
p
[for
sufficiently
small
H
3
],
hence
determines
a
J
1
-module
that
occurs
as
a
subquotient
[not
only
of
the
J
1
-module
H
1
ab-t
⊗
Q
p
,
but
also,
in
light
of
our
assumption
that
w
≥
2!]
of
the
J
1
-module
Q
1
⊗
Q
p
leads
to
a
contradiction
[since
the
J
1
-module
Q
1
⊗
Q
p
is
Hodge-Tate
with
weights
∈
{0,
1}].
Thus,
we
conclude
that
φ
:
G
1
→
G
2
=
G
3
is
of
01-qLT-type,
hence
geometric,
i.e.,
arises
from
a
unique
embedding
of
fields
k
2
→
k
1
of
finite
degree.
Finally,
the
geometricity
of
φ
implies
that
the
geometricity
of
ψ
may
be
derived
from
the
“relative”
result
given
in
[Mzk3],
Theorem
B.
Remark
3.9.1.
The
proof
given
above
of
Corollary
3.9
shows
that
the
“Π
2
”
of
Corollary
3.9
may,
in
fact,
be
taken
to
be
a
“Π
2
”
as
in
Corollary
3.8,
(g).
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
I
45
Section
4:
Chains
of
Elementary
Operations
In
the
present
§4,
we
generalize
[cf.
Theorems
4.7,
4.12;
Remarks
4.7.1,
4.12.1
below]
the
theory
of
“categories
of
dominant
localizations”
discussed
in
[Mzk9],
§2
[cf.
also
the
tempered
versions
of
these
categories,
discussed
in
[Mzk10],
§6],
to
include
“localizations”
obtained
by
more
general
“chains
of
elementary
opera-
tions”
—
i.e.,
the
operations
of
passing
to
a
finite
étale
covering,
passing
to
a
finite
étale
quotient,
“de-cuspidalization”,
and
“de-orbification”
[cf.
Definition
4.2
below;
[Mzk13],
§2]
—
which
are
applied
to
some
given
algebraic
stack
over
a
field.
The
field
and
algebraic
stack
under
consideration
are
quite
general
in
nature
[by
com-
parison,
e.g.,
to
the
theory
of
[Mzk9],
§2;
[Mzk13],
§2],
but
are
subject
to
various
assumptions.
One
key
assumption
asserts
that
the
algebraic
stack
satisfies
a
certain
relative
version
of
the
“Grothendieck
Conjecture”.
Before
proceeding,
we
recall
the
following
immediate
consequence
of
[Mzk13],
Lemma
2.1;
[Mzk12],
Proposition
1.2,
(ii).
Lemma
4.1.
(Decomposition
Groups
of
Hyperbolic
Orbicurves)
Let
Σ
be
a
nonempty
set
of
prime
numbers,
Δ
a
pro-Σ
group
of
GFG-type
that
admits
base-prime
[cf.
Definition
2.1,
(iv)]
partial
construction
data
(k,
X,
Σ)
[consisting
of
the
construction
data
field,
construction
data
base-stack,
and
construction
data
prime
set]
such
that
X
is
a
hyperbolic
orbicurve
[cf.
§0],
and
k
is
algebraically
closed.
Let
x
A
(respectively,
x
B
=
x
A
)
be
either
a
closed
point
or
a
cusp
[cf.
§0]
of
X;
A
⊆
Δ
(respectively,
B
⊆
Δ)
the
decomposition
group
[well-defined
up
to
conjugation
in
Δ]
of
x
A
(respectively,
x
B
).
Then:
(i)
A,
B
are
pro-cyclic
groups;
A
B
=
{1}.
If
x
A
is
a
closed
point
of
X,
and
A
=
{1},
then
A
is
a
finite,
normally
terminal
[cf.
§0]
subgroup
of
Δ.
If
x
A
is
a
cusp,
then
A
is
a
torsion-free,
commensurably
terminal
[cf.
§0]
infinite
subgroup
of
Δ.
(ii)
The
order
of
every
finite
cyclic
closed
subgroup
C
⊆
Δ
divides
the
order
of
X
[cf.
§0].
(iii)
Every
finite
nontrivial
closed
subgroup
C
⊆
Δ
is
contained
in
a
decomposition
group
of
a
unique
closed
point
of
X.
In
particular,
the
non-
trivial
decomposition
groups
of
closed
points
of
X
may
be
characterized
[“group-
theoretically”]
as
the
maximal
finite
nontrivial
closed
subgroups
of
Δ.
(iv)
X
is
a
hyperbolic
curve
if
and
only
if
Δ
is
torsion-free.
(v)
Suppose
that
the
quotient
ψ
A
:
Δ
Δ
A
of
Δ
by
the
closed
normal
sub-
group
of
Δ
topologically
generated
by
A
is
slim
and
nontrivial.
If
x
A
is
a
closed
point
of
X
(respectively,
a
cusp),
then
we
suppose
further
that
Σ
=
Primes
[which
forces
the
characteristic
of
k
to
be
zero]
(respectively,
that
A
⊆
J
for
some
normal
open
torsion-free
subgroup
J
of
Δ).
Then
Δ
A
is
a
profinite
group
of
GFG-type
that
admits
base-prime
partial
construction
data
(k,
X
A
,
Σ)
[consisting
of
the
con-
struction
data
field,
construction
data
base-stack,
and
construction
data
prime
set]
such
that
X
A
is
a
hyperbolic
orbicurve
equipped
with
a
dominant
k-morphism
φ
A
:
X
→
X
A
that
is
uniquely
determined
[up
to
a
unique
isomorphism]
by
the
46
SHINICHI
MOCHIZUKI
property
that
it
induces
[up
to
composition
with
an
inner
automorphism]
ψ
A
.
More-
over,
if
x
A
is
a
closed
point
of
X
(respectively,
a
cusp),
then
φ
A
is
a
partial
coarsification
morphism
[cf.
§0]
which
is
an
isomorphism
either
over
X
A
or
over
the
complement
in
X
A
of
the
point
of
X
A
determined
by
x
A
(respectively,
is
an
open
immersion
whose
image
is
the
complement
of
the
point
of
X
A
determined
by
x
A
).
(vi)
In
the
notation
of
(v),
if
B
=
{1},
then
ψ
A
(B)
=
{1}.
Proof.
First,
we
recall
that
by
the
definition
of
a
profinite
group
of
GFG-type
[cf.
the
discussion
at
the
beginning
of
§2],
it
follows
that
there
exists
a
normal
open
subgroup
H
⊆
Δ
such
that
if
we
write
X
H
→
X
for
the
corresponding
Galois
covering,
then
X
H
is
a
hyperbolic
curve.
Next,
let
us
observe
that,
in
light
of
our
assumption
that
the
partial
construction
data
is
base-prime,
we
may
lift
the
entire
situation
to
characteristic
zero,
hence
assume,
at
least
for
the
proof
of
assertions
(i),
(ii),
(iii),
(iv),
that
k
is
of
characteristic
zero.
Thus,
assertions
(i),
(ii),
(iii)
when
x
A
,
x
B
are
closed
points
(respectively,
cusps)
of
X
follow
immediately
from
[Mzk13],
Lemma
2.1
(respectively,
[Mzk12],
Proposition
1.2,
(ii)).
Next,
we
consider
assertion
(iv).
First,
we
observe
that
the
necessity
portion
of
assertion
(iv)
follows
immediately
from
assertion
(iii).
To
verify
sufficiency,
let
us
suppose
that
Δ
is
torsion-free.
Let
π
1
tame
(X)
Δ
be
a
scheme-theoretic
envelope
of
Δ.
Then
since
X
H
is
a
scheme,
it
follows
that
the
nontrivial
[finite
closed]
subgroups
of
π
1
tame
(X)
that
arise
as
decomposition
groups
of
closed
points
map
injectively,
via
the
composite
surjection
π
1
tame
(X)
Δ
Δ/H,
into
Δ/H,
hence,
a
fortiori,
injectively
via
the
surjection
π
1
tame
(X)
Δ,
into
Δ
[which
is
torsion-free].
Thus,
the
decomposition
groups
in
π
1
(X)
=
π
1
tame
(X)
[cf.
our
assumption
that
k
is
algebraically
closed
of
characteristic
zero]
of
closed
points
of
X
are
trivial.
But
this
implies
[by
considering,
for
instance,
the
Galois
covering
X
H
→
X]
that
X
is
a
scheme,
as
desired.
This
completes
the
proof
of
assertion
(iv).
Next,
we
consider
assertion
(v).
First,
let
us
observe
that
X
A
admits
a
finite
étale
covering
Y
A
→
X
A
arising
from
a
normal
open
subgroup
of
Δ
A
such
that
Y
A
is
a
curve,
which
will
necessarily
be
hyperbolic,
in
light
of
the
slimness
and
nontriviality
of
Δ
A
.
Indeed,
when
x
A
is
a
closed
point
of
X
[so
Σ
=
Primes;
k
is
of
characteristic
zero],
this
follows
immediately
from
the
equivalence
of
definitions
of
a
“hyperbolic
orbicurve”
discussed
in
§0;
when
x
A
is
a
cusp,
this
follows
from
assertion
(iv)
and
our
assumption
of
the
existence
of
the
subgroup
J
⊆
Δ.
Now
the
remainder
of
assertion
(v)
follows
immediately
from
the
definitions.
This
completes
the
proof
of
assertion
(v).
Finally,
we
consider
assertion
(vi).
Assertion
(vi)
is
immediate
if
x
B
is
a
cusp
[cf.
assertion
(i)];
thus,
we
may
assume
that
x
B
is
a
closed
point
of
X.
If
ψ
A
(B)
=
{1},
then
it
follows
that
the
decomposition
group
⊆
Δ
A
of
the
image
of
x
B
in
X
A
is
trivial.
Since
[by
assertion
(v)]
X
A
admits
a
finite
étale
covering
Y
A
→
X
A
arising
from
an
open
subgroup
of
Δ
A
such
that
Y
A
is
a
hyperbolic
curve,
we
thus
conclude
that
X
A
is
scheme-like
in
a
neighborhood
of
the
image
of
x
B
in
X
A
,
hence
[in
light
of
the
explicit
description
of
the
morphism
φ
A
in
the
statement
of
assertion
(v)]
that
X
is
scheme-like
in
a
neighborhood
of
x
B
.
But
this
implies
that
B
=
{1}.
This
completes
the
proof
of
assertion
(vi).
Remark
4.1.1.
Note
that
Lemma
4.1,
(iv),
is
false
if
we
only
assume
that
Δ
is
almost
pro-Σ.
Indeed,
such
an
example
may
be
constructed
by
taking
X
to
be
a
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
I
47
hyperbolic
curve
over
an
algebraically
closed
field
k
of
characteristic
zero,
Y
→
X
a
finite
étale
Galois
covering
of
degree
prime
to
Σ,
and
Δ
to
be
the
quotient
of
π
1
(X)
by
the
kernel
of
the
surjection
(π
1
(X)
⊇)
π
1
(Y
)
π
1
(Y
)
(Σ)
to
the
maximal
pro-Σ
quotient
π
1
(Y
)
(Σ)
of
π
1
(Y
).
Then
for
any
prime
p
dividing
the
order
of
Gal(Y
/X)
[so
p
∈
Σ],
it
follows
by
considering
Sylow
p-subgroups
that
Δ
contains
an
element
of
order
p,
despite
the
fact
that
X
is
a
curve.
Definition
4.2.
Let
G
be
a
slim
profinite
group;
1
→
Δ
→
Π
→
G
→
1
an
extension
of
GSAFG-type
that
admits
base-prime
partial
construction
data
(k,
X,
Σ),
where
Σ
=
∅;
α
:
π
1
tame
(X)
Π
is
a
scheme-theoretic
envelope.
Thus,
if
we
write
π
1
tame
(X)
G
k
for
the
quotient
given
by
the
absolute
Galois
group
G
k
→
X
of
k,
then
α
determines
a
scheme-theoretic
envelope
β
:
G
k
G.
Write
X
for
the
pro-finite
étale
covering
of
X
determined
by
the
surjection
α;
k
for
the
for
the
projective
resulting
field
extension
of
k.
In
a
similar
vein,
we
shall
write
Π
system
of
profinite
groups
determined
by
the
open
subgroups
of
Π.
[Thus,
one
and
a
profinite
group
by
thinking
of
the
may
consider
homomorphisms
between
Π
profinite
group
as
a
trivial
projective
system
of
profinite
groups
—
cf.
the
theory
of
“pro-anabelioids”,
as
in
[Mzk8],
Definition
1.2.6.]
Then:
(i)
We
shall
refer
to
as
an
[
X/X-]chain
[of
length
n]
[where
n
≥
0
is
an
integer]
any
finite
sequence
X
0
X
1
.
.
.
X
n−1
X
n
of
generically
scheme-like
algebraic
stacks
X
j
[for
j
=
0,
.
.
.
,
n],
each
equipped
with
→
X
j
satisfying
the
following
conditions:
a
dominant
“rigidifying
morphism”
ρ
j
:
X
→
X].
(0
X
)
X
0
=
X
[equipped
with
its
natural
rigidifying
morphism
X
(1
X
)
There
exists
a
[uniquely
determined]
morphism
X
j
→
Spec(k
j
)
com-
k
is
a
finite
extension
of
k
such
that
X
j
is
patible
with
ρ
j
,
where
k
j
⊆
geometrically
connected
over
k
j
.
j
→
X
j
such
(2
X
)
Each
ρ
j
determines
a
maximal
pro-finite
étale
covering
X
→
X
j
admits
a
factorization
X
→
X
j
→
X
j
.
The
kernel
Δ
j
of
that
X
the
resulting
natural
surjection
j
/X
j
)
G
j
=
Gal(
Π
j
=
Gal(
X
k/k
j
)
def
def
is
slim
and
nontrivial;
every
prime
dividing
the
order
of
a
finite
quotient
group
of
Δ
j
is
invertible
in
k.
(3
X
)
Suppose
that
X
is
a
hyperbolic
orbicurve
[over
k].
Then
each
X
j
is
also
a
hyperbolic
orbicurve
[over
k
j
].
Moreover,
each
Δ
j
is
a
pro-Σ
group.
(4
X
)
Each
“X
j
X
j+1
”
[for
j
=
0,
.
.
.
,
n
−
1]
is
an
“elementary
operation”,
as
defined
below.
48
SHINICHI
MOCHIZUKI
Here,
an
elementary
operation
“X
j
X
j+1
”
is
defined
to
consist
of
the
datum
of
a
dominant
“operation
morphism”
φ
either
from
X
j
to
X
j+1
or
from
X
j+1
to
X
j
which
is
compatible
with
ρ
j
,
ρ
j+1
,
and,
moreover,
is
of
one
of
the
following
four
types:
(a)
Type
:
In
this
case,
the
elementary
operation
X
j
X
j+1
consists
of
a
finite
étale
covering
φ
:
X
j+1
→
X
j
.
Thus,
φ
determines
an
open
immersion
of
profinite
groups
Π
j+1
→
Π
j
.
(b)
Type
:
In
this
case,
the
elementary
operation
X
j
X
j+1
consists
of
a
finite
étale
morphism
φ
:
X
j
→
X
j+1
—
i.e.,
a
“finite
étale
quotient”.
Thus,
φ
determines
an
open
immersion
of
profinite
groups
Π
j
→
Π
j+1
.
(c)
Type
•:
This
type
of
elementary
operation
is
only
defined
if
X
is
a
hyperbolic
orbicurve.
In
this
case,
the
elementary
operation
X
j
X
j+1
consists
of
an
open
immersion
φ
:
X
j
→
X
j+1
[so
k
j
=
k
j+1
]
—
i.e.,
a
“de-cuspidalization”
—
such
that
the
image
of
φ
is
the
complement
of
a
single
k
j+1
-valued
point
of
X
j+1
whose
decomposition
group
in
Δ
j
is
contained
in
some
normal
open
torsion-free
subgroup
of
Δ
j
.
Thus,
φ
determines
a
surjection
of
profinite
groups
Π
j
Π
j+1
.
(d)
Type
:
This
type
of
elementary
operation
is
only
defined
if
X
is
a
hyperbolic
orbicurve
and
Σ
=
Primes
[which
forces
the
characteristic
of
k
to
be
zero].
In
this
case,
the
elementary
operation
X
j
X
j+1
consists
of
a
partial
coarsification
morphism
[cf.
§0]
φ
:
X
j
→
X
j+1
[so
k
j
=
k
j+1
]
—
i.e.,
a
“de-orbification”
—
such
that
φ
is
an
isomorphism
over
the
complement
in
X
j+1
of
some
k
j+1
-valued
point
of
X
j+1
.
Thus,
φ
determines
a
surjection
of
profinite
groups
Π
j
Π
j+1
.
Thus,
any
X/X-chain
determines
a
sequence
of
symbols
∈
{,
,
•,
}
[correspond-
ing
to
the
types
of
elementary
operations
in
the
X/X-chain],
which
we
shall
refer
to
as
the
type-chain
associated
to
the
X/X-chain.
(ii)
An
isomorphism
between
two
X/X-chains
with
identical
type-chains
[hence
of
the
same
length]
∼
(X
0
.
.
.
X
n
)
→
(Y
0
.
.
.
Y
n
)
is
defined
to
be
a
collection
of
isomorphisms
of
generically
scheme-like
algebraic
∼
stacks
X
j
→
Y
j
[for
j
=
0,
.
.
.
,
n]
that
are
compatible
with
the
rigidifying
mor-
phisms.
[Here,
we
note
that
the
condition
of
compatibility
with
the
rigidifying
morphisms
implies
that
every
automorphism
of
an
X/X-chain
is
given
by
the
iden-
tity,
and
that
every
isomorphism
of
X/X-chains
is
compatible
with
the
respective
operation
morphisms.]
Thus,
one
obtains
a
category
Chain(
X/X)
whose
objects
are
the
X/X-chains
[with
arbitrary
associated
type-chain],
and
whose
morphisms
are
the
isomorphisms
between
X/X-chains
[with
identical
type-chains].
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
I
49
A
terminal
morphism
between
two
X/X-chains
[with
arbitrary
associated
type-
chains]
(X
0
.
.
.
X
n
)
→
(Y
0
.
.
.
Y
m
)
is
defined
to
be
a
dominant
k-morphism
X
n
→
Y
m
.
Thus,
one
obtains
a
category
Chain
trm
(
X/X)
whose
objects
are
the
X/X-chains
[with
arbitrary
associated
type-chain],
and
whose
morphisms
are
the
terminal
morphisms
between
X/X-chains;
write
⊆
Chain
trm
(
X/X)
Chain
iso-trm
(
X/X)
for
the
subcategory
determined
by
the
terminal
isomorphisms
[i.e.,
the
isomor-
phisms
of
Chain
trm
(
X/X)].
Thus,
it
follows
immediately
from
the
definitions
that
we
obtain
natural
functors
Chain(
X/X)
→
Chain
iso-trm
(
X/X)
→
Chain
trm
(
X/X).
(iii)
We
shall
refer
to
as
a
[Π-]chain
[of
length
n]
[where
n
≥
0
is
an
integer]
any
finite
sequence
Π
0
Π
1
.
.
.
Π
n−1
Π
n
of
slim
profinite
groups
Π
j
[for
j
=
0,
.
.
.
,
n],
each
equipped
with
an
open
“rigid-
→
Π
j
[i.e.,
since
we
are
working
with
slim
profinite
ifying
homomorphism”
ρ
j
:
Π
groups,
an
open
homomorphism
from
some
open
subgroup
of
Π
to
Π
j
]
satisfying
the
following
conditions:
→
Π].
(0
Π
)
Π
0
=
Π
[equipped
with
its
natural
rigidifying
homomorphism
Π
(1
Π
)
There
exists
a
[uniquely
determined]
surjection
Π
j
G
j
,
where
G
j
⊆
G
is
an
open
subgroup,
that
is
compatible
with
ρ
j
and
the
natural
composite
→
Π
G.
morphism
Π
(2
Π
)
Each
kernel
def
Δ
j
=
Ker(Π
j
G
j
→
G)
is
slim
and
nontrivial;
every
prime
dividing
the
order
of
a
finite
quotient
group
of
Δ
j
is
invertible
in
k.
(3
Π
)
Suppose
that
X
is
a
hyperbolic
orbicurve
[over
k].
Then
each
Δ
j
is
a
pro-Σ
group.
Also,
we
shall
refer
to
as
a
cuspidal
decomposition
group
in
Δ
j
any
commensurator
in
Δ
j
of
a
nontrivial
image
via
ρ
j
of
the
inverse
of
the
decomposition
group
in
Δ
[determined
by
α]
of
a
cusp
image
in
Π
of
X.
(4
Π
)
Each
“Π
j
Π
j+1
”
[for
j
=
0,
.
.
.
,
n
−
1]
is
an
“elementary
operation”,
as
defined
below.
Here,
an
elementary
operation
“Π
j
Π
j+1
”
is
defined
to
consist
of
the
datum
of
an
open
“operation
homomorphism”
φ
either
from
Π
j
to
Π
j+1
or
from
Π
j+1
to
Π
j
which
is
compatible
with
ρ
j
,
ρ
j+1
,
and,
moreover,
is
of
one
of
the
following
four
types:
50
SHINICHI
MOCHIZUKI
(a)
Type
:
In
this
case,
the
elementary
operation
Π
j
Π
j+1
consists
of
an
open
immersion
of
profinite
groups
φ
:
Π
j+1
→
Π
j
.
(b)
Type
:
In
this
case,
the
elementary
operation
Π
j
Π
j+1
consists
of
an
open
immersion
of
profinite
groups
φ
:
Π
j
→
Π
j+1
.
(c)
Type
•:
This
type
of
elementary
operation
is
only
defined
if
X
is
a
hyperbolic
orbicurve.
In
this
case,
the
elementary
operation
Π
j
Π
j+1
consists
of
a
surjection
of
profinite
groups
φ
:
Π
j
Π
j+1
,
such
that
Ker(φ)
is
topologically
normally
generated
by
a
cuspidal
decomposition
group
C
in
Δ
j
such
that
C
is
contained
in
some
normal
open
torsion-free
subgroup
of
Δ
j
.
(d)
Type
:
This
type
of
elementary
operation
is
only
defined
if
X
is
a
hyperbolic
orbicurve
and
Σ
=
Primes
[which
forces
the
characteristic
of
k
to
be
zero].
In
this
case,
the
elementary
operation
Π
j
Π
j+1
consists
of
a
surjection
of
profinite
groups
φ
:
Π
j
Π
j+1
,
such
that
Ker(φ)
is
topologically
normally
generated
by
a
finite
closed
subgroup
of
Δ
j
.
Thus,
any
Π-chain
determines
a
sequence
of
symbols
∈
{,
,
•,
}
[corresponding
to
the
types
of
elementary
operations
in
the
Π-chain],
which
we
shall
refer
to
as
the
type-chain
associated
to
the
Π-chain.
(iv)
An
isomorphism
between
two
Π-chains
with
identical
type-chains
[hence
of
the
same
length]
∼
(Π
0
.
.
.
Π
n
)
→
(Ψ
0
.
.
.
Ψ
n
)
∼
is
defined
to
be
a
collection
of
isomorphisms
of
profinite
groups
Π
j
→
Ψ
j
[for
j
=
0,
.
.
.
,
n]
that
are
compatible
with
the
rigidifying
homomorphisms.
[Here,
we
note
that
the
condition
of
compatibility
with
the
rigidifying
homomorphisms
implies
[since
all
of
the
profinite
groups
involved
are
slim]
that
every
automorphism
of
a
Π-
chain
is
given
by
the
identity,
and
that
every
isomorphism
of
Π-chains
is
compatible
with
the
respective
operation
homomorphisms.]
Thus,
one
obtains
a
category
Chain(Π)
whose
objects
are
the
Π-chains
[with
arbitrary
associated
type-chain],
and
whose
morphisms
are
the
isomorphisms
between
Π-chains
[with
identical
type-chains].
A
terminal
homomorphism
between
two
Π-chains
[with
arbitrary
associated
type-
chains]
(Π
0
.
.
.
Π
n
)
→
(Ψ
0
.
.
.
Ψ
m
)
is
defined
to
be
an
open
outer
homomorphism
Π
n
→
Ψ
m
that
is
compatible
[up
to
composition
with
an
inner
automorphism]
with
the
open
homomorphisms
Π
n
→
G,
Ψ
m
→
G.
Thus,
one
obtains
a
category
Chain
trm
(Π)
whose
objects
are
the
Π-chains
[with
arbitrary
associated
type-chain],
and
whose
morphisms
are
the
terminal
homomorphisms
between
Π-chains;
write
Chain
iso-trm
(Π)
⊆
Chain
trm
(Π)
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
I
51
for
the
subcategory
determined
by
the
terminal
isomorphisms
[i.e.,
the
isomor-
phisms
of
Chain
trm
(Π)].
Thus,
it
follows
immediately
from
the
definitions
that
we
obtain
natural
functors
Chain(Π)
→
Chain
iso-trm
(Π)
→
Chain
trm
(Π).
(v)
We
shall
use
the
notation
Chain
iso-trm
(∼){−}
⊆
Chain
iso-trm
(∼);
Chain
trm
(∼){−}
⊆
Chain
trm
(∼)
—
where
“(∼)”
is
either
equal
to
“(
X/X)”
or
“(Π)”,
and
“{−}”
contains
some
subset
of
the
set
of
symbols
{,
,
•,
}
—
to
denote
the
respective
full
subcategories
determined
by
the
chains
whose
associated
type-chain
only
contains
the
symbols
that
belong
to
“{−}”.
In
particular,
we
shall
write:
DLoc(
X/X)
=
Chain
trm
(
X/X){,
•};
def
ÉtLoc(
X/X)
=
Chain
iso-trm
(
X/X){,
};
def
def
DLoc(Π)
=
Chain
trm
(Π){,
•}
def
ÉtLoc(Π)
=
Chain
iso-trm
(Π){,
}
[cf.
the
theory
of
[Mzk9],
§2;
Remark
4.7.1
below].
Remark
4.2.1.
Thus,
it
follows
immediately
from
the
definitions
that
if,
in
the
notation
of
Definition
4.2,
(i),
X
0
X
1
.
.
.
X
n−1
X
n
is
an
X/X-chain,
then
the
resulting
profinite
groups
Π
j
determine
a
Π-chain
Π
0
Π
1
.
.
.
Π
n−1
Π
n
with
the
same
associated
type-chain.
In
particular,
we
obtain
natural
functors
Chain(
X/X)
→
Chain(Π)
Chain
iso-trm
(
X/X)
→
Chain
iso-trm
(Π);
Chain
trm
(
X/X)
→
Chain
trm
(Π)
which
are
compatible
with
the
natural
functors
of
Definition
4.2,
(ii),
(iv).
Remark
4.2.2.
Note
that
in
the
situation
of
Definition
4.2,
(i),
G
j
is
a
slim
profinite
group;
1
→
Δ
j
→
Π
j
→
G
j
→
1
is
an
extension
of
GSAFG-type
that
admits
base-prime
partial
construction
data
(k
j
,
X
j
,
Σ),
where
X
j
is
a
hyperbolic
orbicurve
whenever
X
0
is
a
hyperbolic
orbicurve;
α,
ρ
j
determine
[in
light
of
the
slimness
of
Π
j
]
a
scheme-theoretic
envelope
α
j
:
π
1
tame
(X
j
)
Π
j
.
That
is
to
say,
we
obtain,
for
each
j,
similar
data
to
the
data
introduced
at
the
beginning
of
Definition
4.2.
Here,
relative
to
issue
of
verifying
that
Δ
j
admits
an
open
subgroup
that
corresponds
to
a
scheme-like
covering
of
X
j
,
it
is
useful
to
recall,
in
the
case
of
de-cuspidalization
operations,
i.e.,
“•”,
the
condition
[cf.
Definition
4.2,
(i),
(c);
Definition
4.2,
(iii),
(c)]
that
the
cuspidal
decomposition
group
under
consideration
be
contained
in
a
normal
open
torsion-free
subgroup
[cf.
Lemma
4.1,
(iv)];
in
the
case
of
de-orbification
operations,
i.e.,
“”,
it
is
useful
to
recall
the
assumption
that
Σ
=
Primes,
together
with
the
equivalence
of
definitions
of
the
notion
of
a
“hyperbolic
orbicurve”
discussed
in
§0.
52
SHINICHI
MOCHIZUKI
Proposition
4.3.
(Re-ordering
of
Chains)
In
the
notation
of
Definition
4.2,
suppose
that
Σ
=
Primes;
let
X
0
.
.
.
X
n
be
an
X/X-chain.
Then
there
exists
a
terminally
isomorphic
X/X-chain
Y
0
.
.
.
Y
m
whose
associated
type-chain
is
of
the
form
,
•,
•,
.
.
.
,
•,
,
,
,
.
.
.
,
,
—
i.e.,
consists
of
the
symbol
,
followed
by
a
sequence
of
the
symbols
•,
followed
by
the
symbol
,
followed
by
a
sequence
of
the
symbols
,
followed
by
the
symbol
.
Moreover,
Y
m−1
→
Y
m
may
be
taken
to
arise
from
an
extension
of
the
base
field
[where
we
recall
that
this
base
field
will
always
be
a
finite
extension
of
k].
Proof.
Indeed,
let
us
first
observe
that
by
taking
Y
m−1
→
Y
m
to
arise
from
an
appropriate
extension
of
the
base
field,
we
may
ignore
the
“k
j
-rationality”
issues
that
occur
in
Definition
4.2,
(i),
(c),
(d).
Next,
let
us
observe
that
it
is
immediate
from
the
definitions
that
we
may
always
“move
the
symbol
to
the
top
of
the
type-
chain”.
This
completes
the
proof
of
Proposition
4.3
when
X
is
not
a
hyperbolic
orbicurve.
Thus,
in
the
remainder
of
the
proof,
we
may
assume
without
loss
of
generality
that
X
is
a
hyperbolic
orbicurve,
and
that
the
symbols
indexed
by
j
≥
1
of
the
type-chain
are
∈
{,
•,
}.
Next,
let
us
observe
that
the
operation
morphisms
indexed
by
j
≥
1
always
have
domain
indexed
by
j
and
codomain
indexed
by
j
+
1.
Thus,
by
composing
these
operation
morphisms,
we
obtain
a
morphism
X
1
→
X
n−1
.
Here,
we
may
assume,
without
loss
of
generality,
that
X
1
is
a
hyperbolic
curve,
and
that
X
1
→
X
n−1
induces
a
Galois
extension
of
function
fields
and
an
isomorphism
of
base
fields.
Also,
we
may
assume
that
the
morphism
X
1
→
X
n−1
factors
through
a
connected
finite
étale
covering
Z
→
X
n−1
,
where
Z
is
a
hyperbolic
curve.
Thus,
by
considering
the
extension
of
function
fields
determined
by
X
1
→
X
n−1
,
it
follows
immediately
that
X
n−1
may
be
obtained
from
X
1
by
applying
de-cuspidalization
operations
[i.e.,
“•”]
to
X
1
at
the
cusps
of
X
1
that
map
to
points
of
X
n−1
,
then
forming
the
stack-theoretic
quotient
by
the
action
of
Gal(X
1
/X
n−1
)
[i.e.,
“”],
and
finally
applying
suitable
de-orbification
[i.e.,
“”]
operations
to
this
quotient
to
recover
X
n−1
.
This
yields
a
type-chain
of
the
desired
form.
As
the
following
example
shows,
the
issue
of
permuting
the
symbols
“”,
“”
is
not
so
straightforward.
Example
4.4.
Non-permutability
of
Étale
Quotients
and
De-orbifications.
In
the
notation
of
Definition
4.2,
let
us
assume
further
Σ
=
Primes
[so
k
is
of
char-
acteristic
zero].
Then
there
exists
an
X/X-chain
X
0
X
1
X
2
of
length
2
with
associated
type-chain
∗
0
,
∗
1
,
where
∗
0
,
∗
1
∈
{,
},
∗
0
=
∗
1
,
which
is
not
terminally
isomorphic
to
any
X/X-chain
Y
0
Y
1
Y
2
of
length
2
with
associated
type-chain
∗
1
,
∗
0
.
Indeed:
(i)
The
case
of
type-chain
,
:
Let
X
be
a
hyperbolic
curve
of
type
(g,
r)
over
k
equipped
with
an
automorphism
σ
of
the
k-scheme
X
of
order
2
that
has
precisely
one
fixed
point
x
∈
X(k);
X
0
=
X
X
1
the
elementary
operation
of
type
given
by
forming
the
stack-theoretic
quotient
of
X
by
the
action
of
σ;
x
1
∈
X
1
(k)
the
image
of
x
in
X
1
;
X
1
X
2
the
nontrivial
elementary
operation
of
type
[i.e.,
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
I
53
such
that
the
corresponding
operation
morphism
X
1
→
X
2
is
a
non-isomorphism]
determined
by
the
point
x
1
∈
X
1
(k).
Thus,
we
assume
that
X
2
is
a
hyperbolic
curve,
whose
type
we
denote
by
(g
2
,
r
2
).
On
the
other
hand,
since
X
is
a
scheme,
any
∼
chain
Y
0
Y
1
Y
2
of
length
2
with
associated
type-chain
,
satisfies
Y
0
→
Y
1
∼
Thus,
if
Y
2
→
[compatibly
with
X].
X
2
over
k,
then
the
covering
X
=
X
0
→
X
2
,
∼
∼
which
is
ramified,
of
degree
2,
together
with
the
covering
X
→
Y
0
→
Y
1
→
Y
2
,
which
is
unramified,
of
some
degree
d,
yields
equations
2
·
χ
2
+
1
=
χ
=
d
·
χ
2
def
def
[where
we
write
χ
=
2g
−
2
+
r,
χ
2
=
2g
2
−
2
+
r
2
]
—
which
imply
[since
d,
χ,
χ
2
are
positive
integers]
that
d
−
2
=
χ
2
=
1,
hence
that
d
=
3,
χ
2
=
1,
χ
=
3.
In
particular,
by
choosing
X
so
that
χ
is
>
3
[e.g.,
X
such
that
g
≥
3],
we
obtain
a
contradiction.
(ii)
The
case
of
type-chain
,
:
Let
X
be
a
proper
hyperbolic
orbicurve
over
k;
X
→
C
the
coarse
space
associated
to
the
algebraic
stack
X.
Let
us
assume
further
that
C
is
a
[proper]
hyperbolic
curve
over
k;
that
the
morphism
X
→
C
is
a
non-isomorphism
which
restricts
to
an
isomorphism
away
from
some
point
c
∈
C(k);
and
that
there
exists
a
finite
étale
covering
:
C
→
D
of
degree
2
[so
D
is
also
a
proper
hyperbolic
curve
over
k,
which
is
not
isomorphic
to
C].
[It
is
easy
to
construct
such
objects
by
starting
from
D
and
then
constructing
C,
X.]
Now
def
we
take
X
0
=
X
X
1
=
C
to
be
the
elementary
operation
of
type
determined
def
by
the
unique
point
of
x
∈
X(k)
lying
over
c
∈
C(k);
C
=
X
1
X
2
=
D
to
be
the
elementary
operation
of
type
determined
by
the
finite
étale
covering
:
C
→
D.
Write
e
x
≥
2
for
the
ramification
index
of
X
→
C
at
x;
g
D
≥
2
for
the
genus
of
def
D;
χ
D
=
2g
D
−
2
≥
2.
On
the
other
hand,
let
us
suppose
that
Y
0
Y
1
Y
2
∼
is
a
chain
of
length
2
with
associated
type-chain
,
such
that
X
2
→
Y
2
over
k.
∼
Then
since
D
=
X
2
→
Y
2
is
a
scheme,
it
follows
that
the
hyperbolic
orbicurve
Y
1
admits
a
point
y
1
∈
Y
1
(k)
such
that
Y
1
is
a
scheme
away
from
y
1
.
Write
e
y
1
for
the
ramification
index
of
the
operation
morphism
Y
1
→
Y
2
at
y
1
.
Note
that
if
Y
1
is
a
scheme,
then
the
finite
étale
covering
X
=
Y
0
of
Y
1
is
as
well
—
a
contradiction.
Thus,
we
conclude
that
Y
1
is
not
a
scheme
at
y
1
,
i.e.,
e
y
1
≥
2.
Next,
let
us
observe
that
if
the
finite
étale
morphism
Y
0
→
Y
1
is
not
an
isomorphism,
i.e.,
of
degree
d
≥
2,
then
the
morphisms
X
→
C
→
D
and
X
=
Y
0
→
Y
1
give
rise
to
a
relation
2χ
D
+
(e
x
−
1)/e
x
=
d(χ
D
+
(e
y
1
−
1)/e
y
1
)
—
i.e.,
1
>
(e
x
−
1)/e
x
=
(d
−
2)χ
D
+
d(e
y
1
−
1)/e
y
1
≥
d(e
y
1
−
1)/e
y
1
≥
d/2
≥
1,
a
contradiction.
Thus,
we
conclude
that
d
=
1,
i.e.,
that
the
operation
morphism
X
=
Y
0
→
Y
1
is
an
isomorphism.
But
this
implies
that
Y
2
is
isomorphic
to
the
∼
coarse
space
associated
to
X,
i.e.,
that
we
have
an
isomorphism
Y
2
→
C,
hence
an
∼
∼
isomorphism
D
=
X
2
→
Y
2
→
C
—
a
contradiction.
Next,
we
recall
the
group-theoretic
characterization
of
the
cuspidal
decomposi-
tion
groups
of
a
hyperbolic
[orbi]curve
given
in
[Mzk12].
Lemma
4.5.
group;
(Cuspidal
Decomposition
Groups)
Let
G
be
a
slim
profinite
1
→
Δ
→
Π
→
G
→
1
54
SHINICHI
MOCHIZUKI
an
extension
of
GSAFG-type
that
admits
base-prime
[cf.
Definition
2.1,
(iv)]
partial
construction
data
(k,
k,
X,
Σ),
where
X
is
a
hyperbolic
orbicurve;
α
:
tame
π
1
(X)
Π
a
scheme-theoretic
envelope;
l
∈
Σ
a
prime
such
that
the
:
G
→
Z
×
cyclotomic
character
χ
cyclo
G
l
[i.e.,
the
character
whose
restriction
to
π
1
tame
(X)
via
α
and
the
surjection
Π
G
is
the
usual
cyclotomic
character
π
1
tame
(X)
Gal(
k/k)
→
Z
×
l
]
has
open
image
[i.e.,
in
the
terminology
of
[Mzk12],
“the
outer
action
of
G
on
Δ
is
l-cyclotomically
full”].
We
recall
from
[Mzk12]
that
a
character
χ
:
G
→
Z
×
l
is
called
Q-cyclotomic
[of
weight
w
∈
Q]
if
there
exist
integers
a,
b,
where
b
>
0,
such
that
χ
b
=
(χ
cyclo
)
a
,
w
=
2a/b
[cf.
[Mzk12],
G
Definition
2.3,
(i),
(ii)].
Then:
(i)
X
is
non-proper
if
and
only
if
every
torsion-free
pro-Σ
open
subgroup
of
Δ
is
free
pro-Σ.
(ii)
Let
M
be
a
finite-dimensional
Q
l
-vector
space
equipped
with
a
continuous
G-action.
Then
we
shall
say
that
this
action
is
quasi-trivial
if
it
factors
through
a
finite
quotient
of
G
[cf.
[Mzk12],
Definition
2.3,
(i)].
We
shall
write
τ
(M
)
for
the
quasi-trivial
rank
of
M
[cf.
[Mzk12],
Definition
2.3,
(i)],
i.e.,
the
sum
of
the
Q
l
-dimensions
of
the
quasi-trivial
subquotients
M
j
/M
j+1
of
any
filtration
M
n
⊆
.
.
.
⊆
M
j
⊆
.
.
.
M
0
=
M
of
M
by
Q
l
[G]-modules
such
that
each
M
j
/M
j+1
is
either
quasi-trivial
or
has
no
nontrivial
subquotients.
If
χ
:
G
→
Z
×
l
is
a
character,
then
we
shall
write
d
χ
(M
)
=
τ
(M
(χ
−1
))
−
τ
(Hom
Q
l
(M,
Q
l
))
def
[where
“M
(χ
−1
)”
denotes
the
result
of
“twisting”
M
by
the
character
χ
−1
].
We
shall
say
that
two
characters
G
→
Z
×
l
are
power-equivalent
if
there
exists
a
positive
integer
n
such
that
the
n-th
powers
of
the
two
characters
coincide.
Then
d
χ
(M
),
regarded
as
a
function
of
χ,
depends
only
on
the
power-equivalence
class
of
χ.
(iii)
Suppose
that
X
is
not
proper
[cf.
(i)].
Then
the
character
G
→
Z
×
l
arising
from
the
determinant
of
the
G-module
H
ab
⊗Q
l
,
where
H
⊆
Δ
is
a
torsion-
free
pro-Σ
characteristic
open
subgroup
such
that
H
ab
⊗
Q
l
=
0,
is
Q-cyclotomic
of
positive
weight.
Moreover,
for
every
sufficiently
small
characteristic
open
subgroup
H
⊆
Δ,
the
power-equivalence
class
of
the
cyclotomic
character
χ
cyclo
G
may
be
characterized
as
the
unique
power-equivalence
class
of
characters
χ
:
G
→
×
×
∗
∗
Z
×
l
of
the
form
χ
=
χ
·
χ
∗
,
where
χ
:
G
→
Z
l
(respectively,
χ
∗
:
G
→
Z
l
)
is
a
Q-cyclotomic
character
χ
•
of
maximal
(respectively,
minimal)
weight
such
ab
that
τ
(M
(χ
−1
⊗
Q
l
)
⊕
Q
l
[where
•
))
=
0
for
some
subquotient
G-module
M
of
(H
the
final
direct
summand
Q
l
is
equipped
with
the
trivial
G-action].
Moreover,
in
this
situation,
if
χ
=
χ
cyclo
,
then
the
divisor
of
cusps
of
the
covering
of
X
×
k
k
G
ab
k).
determined
by
H
is
a
disjoint
union
of
d
χ
(H
⊗
Q
l
)
+
1
copies
of
Spec(
(iv)
Suppose
that
X
is
not
proper
[cf.
(i)].
Let
H
⊆
Δ
be
a
torsion-free
pro-Σ
characteristic
open
subgroup;
H
H
∗
the
maximal
pro-l
quotient
of
H.
Then
the
decomposition
groups
of
cusps
⊆
H
∗
may
be
characterized
[“group-
theoretically”]
as
the
maximal
closed
subgroups
I
⊆
H
∗
isomorphic
to
Z
l
which
satisfy
the
following
condition:
We
have
d
χ
cyclo
(J
ab
⊗
Q
l
)
+
1
=
[I
·
J
:
J]
·
d
χ
cyclo
((I
·
J)
ab
⊗
Q
l
)
+
1
G
G
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
I
55
[i.e.,
“the
covering
of
curves
corresponding
to
J
⊆
I
·
J
is
totally
ramified
at
precisely
one
cusp”]
for
every
characteristic
open
subgroup
J
⊆
H
∗
.
(v)
Let
X,
H,
H
∗
be
as
in
(iv).
Then
the
set
of
cusps
of
the
covering
of
X
×
k
k
determined
by
H
is
in
natural
bijective
correspondence
with
the
set
of
conjugacy
classes
in
H
∗
of
decomposition
groups
of
cusps
[as
described
in
(iv)].
Moreover,
this
correspondence
is
functorial
in
H
and
compatible
with
the
natural
actions
by
Π
on
both
sides.
In
particular,
by
allowing
H
to
vary,
this
yields
a
[“group-
theoretic”]
characterization
of
the
decomposition
groups
of
cusps
in
Π.
(vi)
Let
I
⊆
Π
be
a
decomposition
group
of
a
cusp.
Then
I
=
C
Π
(I
Δ)
[cf.
§0].
Proof.
Assertion
(i)
may
be
reduced
to
the
case
of
hyperbolic
curves
via
Lemma
4.1,
(iv),
in
which
case
it
is
well-known
[cf.,
e.g.,
[Mzk12],
Remark
1.1.3].
Assertion
(ii)
is
immediate
from
the
definitions.
Assertion
(iii)
follows
immediately
from
[Mzk12],
Proposition
2.4,
(iv),
(vii);
the
proof
of
[Mzk12],
Corollary
2.7,
(i).
Assertion
(iv)
is
[in
light
of
assertion
(iii)]
precisely
a
summary
of
the
argument
of
[Mzk12],
Theorem
1.6,
(i).
Finally,
assertions
(v),
(vi)
follow
immediately
from
[Mzk12],
Proposition
1.2,
(i),
(ii).
Definition
4.6.
(i)
Let
V
(respectively,
F;
S)
be
a
set
of
isomorphism
classes
of
algebraic
stacks
(respectively,
set
of
isomorphism
classes
of
fields;
set
of
nonempty
subsets
of
Primes);
D
⊆
V
×
F
×
S
a
subset
of
the
direct
product
set
V
×
F
×
S,
which
we
shall
think
of
as
a
set
of
collections
of
partial
construction
data.
In
the
following
discussion,
we
shall
use
“[−]”
to
denote
the
isomorphism
class
of
“−”.
We
shall
say
that
D
is
chain-full
if
for
every
extension
1
→
Δ
→
Π
→
G
→
1
of
GSAFG-type,
where
G
is
slim,
that
admits
base-prime
partial
construction
data
(X,
k,
Σ)
such
that
([X],
[k],
Σ)
∈
D
[cf.
Definition
4.2],
it
follows
that
every
“X
j
,
k
j
”
[cf.
Definition
4.2,
(i)]
appearing
in
→
X
is
the
pro-finite
étale
covering
of
X
determined
by
an
X/X-chain
[where
X
some
scheme-theoretic
envelope
for
Π]
determines
an
element
([X
j
],
[k
j
],
Σ)
∈
D.
(ii)
Let
D
be
as
in
(i);
suppose
that
D
is
chain-full.
Then
we
shall
say
that
the
rel-isom-DGC
holds
[i.e.,
“the
relative
isomorphism
version
of
the
Grothendieck
Conjecture
for
D
holds”]
(respectively,
the
rel-hom-DGC
holds
[i.e.,
“the
relative
homomorphism
version
of
the
Grothendieck
Conjecture
for
D
holds”]),
or
that,
the
rel-isom-GC
holds
for
D
(respectively,
the
rel-hom-GC
holds
for
D)
if
the
following
condition
is
satisfied:
For
i
=
1,
2,
let
1
→
Δ
i
→
Π
i
→
G
i
→
1
be
an
extension
of
GSAFG-type,
where
G
i
is
slim,
that
admits
base-prime
partial
construction
data
(k
i
,
X
i
,
Σ
i
)
such
that
([X
i
],
[k
i
],
Σ
i
)
∈
D;
α
i
:
π
1
tame
(X
i
)
Π
i
a
56
SHINICHI
MOCHIZUKI
∼
scheme-theoretic
envelope;
ζ
k
:
k
1
→
k
2
an
isomorphism
of
fields
that
induces,
via
∼
the
α
i
,
an
outer
isomorphism
ζ
G
:
G
1
→
G
2
.
Then
the
natural
map
Isom
k
1
,k
2
(X
1
,
X
2
)
→
Isom
out
G
1
,G
2
(Π
1
,
Π
2
)
out-open
(respectively,
Hom
dom
k
1
,k
2
(X
1
,
X
2
)
→
Hom
G
1
,G
2
(Π
1
,
Π
2
))
∼
determined
by
the
α
i
from
the
set
of
isomorphisms
of
schemes
X
1
→
X
2
lying
over
∼
ζ
k
:
k
1
→
k
2
(respectively,
the
set
of
dominant
morphisms
of
schemes
X
1
→
X
2
∼
∼
lying
over
ζ
k
:
k
1
→
k
2
)
to
the
set
of
outer
isomorphisms
of
profinite
groups
Π
1
→
Π
2
∼
lying
over
ζ
G
:
G
1
→
G
2
(respectively,
the
set
of
open
outer
homomorphisms
of
∼
profinite
groups
Π
1
→
Π
2
lying
over
ζ
G
:
G
1
→
G
2
)
is
a
bijection.
Remark
4.6.1.
Of
course,
in
a
similar
vein,
one
may
also
formulate
the
notions
that
“the
absolute
isomorphism
version
of
the
Grothendieck
Conjecture
holds
for
D”,
“the
absolute
homomorphism
version
of
the
Grothendieck
Conjecture
holds
for
D”,
“the
semi-absolute
isomorphism
version
of
the
Grothendieck
Conjecture
holds
for
D”,
“the
semi-absolute
homomorphism
version
of
the
Grothendieck
Conjecture
holds
for
D”,
etc.
Since
we
shall
not
use
these
versions
in
the
discussion
to
follow,
we
leave
the
routine
details
of
their
formulation
to
the
interested
reader.
Theorem
4.7.
(Semi-absoluteness
of
Chains
of
Elementary
Operations)
Let
D
be
a
chain-full
set
of
collections
of
partial
construction
data
[cf.
Def-
inition
4.6,
(i)]
such
that
the
rel-isom-DGC
holds
[cf.
Definition
4.6,
(ii)].
For
i
=
1,
2,
let
G
i
be
a
slim
profinite
group;
1
→
Δ
i
→
Π
i
→
G
i
→
1
an
extension
of
GSAFG-type
that
admits
base-prime
[cf.
Definition
2.1,
(iv)]
partial
construction
data
(k
i
,
k
i
,
X
i
,
Σ
i
)
such
that
([X
i
],
[k
i
],
Σ
i
)
∈
D;
α
i
:
tame
π
1
(X
i
)
Π
i
a
scheme-theoretic
envelope.
Also,
let
us
suppose
further
that
the
following
conditions
are
satisfied:
(a)
if
either
X
1
or
X
2
is
a
hyperbolic
orbicurve,
then
both
X
1
and
X
2
are
hyperbolic
orbicurves;
(b)
if
either
X
1
or
X
2
is
a
non-proper
hyperbolic
orbicurve,
then
there
exists
a
prime
number
l
∈
Σ
1
Σ
2
such
that
for
i
=
1,
2,
the
cyclotomic
tame
(X
i
)
character
G
i
→
Z
×
l
[i.e.,
the
character
whose
restriction
to
π
1
via
α
i
and
the
surjection
Π
i
G
i
is
the
usual
cyclotomic
character
π
1
tame
(X
i
)
Gal(
k
i
/k
i
)
→
Z
×
l
]
has
open
image.
Let
∼
φ
:
Π
1
→
Π
2
∼
be
an
isomorphism
of
profinite
groups
that
induces
isomorphisms
φ
Δ
:
Δ
1
→
Δ
2
,
∼
φ
G
:
G
1
→
G
2
.
Then:
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
I
57
(i)
The
natural
functors
[cf.
Remark
4.2.1]
i
/X
i
)
→
Chain(Π
i
);
Chain(
X
i
/X
i
)
→
Chain
iso-trm
(Π
i
)
Chain
iso-trm
(
X
i
/X
i
)
→
ÉtLoc(Π
i
)
ÉtLoc(
X
are
equivalences
of
categories
that
are
compatible
with
passing
to
type-chains.
(ii)
The
isomorphism
φ
induces
equivalences
of
categories
∼
Chain(Π
1
)
→
Chain(Π
2
);
∼
Chain
iso-trm
(Π
1
)
→
Chain
iso-trm
(Π
2
)
∼
ÉtLoc(Π
1
)
→
ÉtLoc(Π
2
)
that
are
compatible
with
passing
to
type-chains
and
functorial
in
φ.
(iii)
Suppose
further
that
the
rel-hom-DGC
holds
[cf.
Definition
4.6,
(ii)],
and
that
for
i
=
1,
2,
X
i
is
a
hyperbolic
orbicurve.
Then
the
natural
functors
[cf.
Remark
4.2.1]
i
/X
i
)
→
Chain
trm
(Π
i
);
Chain
trm
(
X
i
/X
i
)
→
DLoc(Π
i
)
DLoc(
X
are
equivalences
of
categories
that
are
compatible
with
passing
to
type-chains.
(iv)
In
the
situation
of
(iii),
the
isomorphism
φ
induces
equivalences
of
cat-
egories
∼
Chain
trm
(Π
1
)
→
Chain
trm
(Π
2
);
∼
DLoc(Π
1
)
→
DLoc(Π
2
)
that
are
compatible
with
passing
to
type-chains
and
functorial
in
φ.
Proof.
First,
we
consider
the
natural
functor
i
/X
i
)
→
Chain(Π
i
)
Chain(
X
of
Remark
4.2.1.
To
conclude
that
this
functor
is
an
equivalence
of
categories,
it
follows
immediately
from
the
definitions
of
the
categories
involved
that
it
suffices
to
i
/X
i
-chain
and
Π
i
-chain
versions
of
the
four
types
of
elementary
verify
that
the
X
operations
,
,
•,
described
in
Definition
4.2,
(i),
(iii),
correspond
bijectively
to
one
another.
This
is
immediate
from
the
definitions
(respectively,
the
cuspidal
portion
of
Lemma
4.1,
(i),
(v);
the
“closed
point
of
X”
portion
of
Lemma
4.1,
(iii),
(v))
for
(respectively,
•;
).
[Here,
we
note
that
in
the
case
of
•,
,
the
“k
j+1
-
rationality”
[in
the
notation
of
Definition
4.2,
(i),
(c),
(d)]
of
the
cusp
or
possibly
non-scheme-like
point
in
question
follows
immediately
from
Lemma
4.1,
(vi),
by
taking
“x
B
”
to
be
the
various
Galois
conjugates
of
this
point.]
Finally,
the
desired
correspondence
for
follows
from
our
assumption
that
the
rel-isom-DGC
holds
by
applying
this
“rel-isom-DGC”
as
was
done
in
the
proofs
of
[Mzk7],
Theorem
2.4;
[Mzk9],
Theorem
2.3,
(i).
This
completes
the
proof
that
the
natural
functor
i
/X
i
)
→
Chain(Π
i
)
is
an
equivalence.
A
similar
application
of
the
“rel-
Chain(
X
∼
i
/X
i
)
→
isom-DGC”
then
yields
the
equivalences
Chain
iso-trm
(
X
Chain
iso-trm
(Π
i
),
58
SHINICHI
MOCHIZUKI
∼
i
/X
i
)
→
ÉtLoc(Π
i
).
In
a
similar
vein,
the
“rel-hom-DGC”
[cf.
asser-
ÉtLoc(
X
∼
i
/X
i
)
→
tion
(iii)]
implies
the
equivalences
of
categories
Chain
trm
(
X
Chain
trm
(Π
i
),
∼
i
/X
i
)
→
DLoc(Π
i
)
of
assertion
(iii).
This
completes
the
proof
of
assertions
DLoc(
X
(i),
(iii).
Finally,
to
obtain
the
equivalences
of
assertions
(ii),
(iv),
it
suffices
to
observe
that
the
definitions
of
the
various
categories
involved
are
entirely
“group-theoretic”.
Here,
we
note
that
the
“group-theoreticity”
of
the
elementary
operations
of
type
,
,
is
immediate;
the
“group-theoreticity”
of
the
elementary
operations
of
type
•
follows
immediately
from
Lemma
4.5,
(v)
[in
light
of
our
assumptions
(a),
(b)].
Also,
we
observe
that
whenever
the
X
i
[for
i
=
1,
2]
are
hyperbolic
orbicurves,
Σ
i
may
be
recovered
“group-theoretically”
from
Δ
i
[i.e.,
as
the
unique
minimal
subset
Σ
⊆
Primes
such
that
Δ
i
is
almost
pro-Σ
].
This
completes
the
proof
of
assertions
(ii),
(iv).
Remark
4.7.1.
The
portion
of
Theorem
4.7
concerning
the
categories
“ÉtLoc(−)”
[cf.
also
Example
4.8
below;
Corollary
2.8,
(ii)]
and
“DLoc(−)”
allows
one
to
relate
the
theory
of
the
present
§4
to
the
theory
of
[Mzk9],
§2
[cf.,
especially,
[Mzk9],
Theorem
2.3].
Example
4.8.
Hyperbolic
Orbicurves.
Let
p
be
a
prime
number;
S
the
set
of
subsets
of
Primes
containing
p;
V
the
set
of
isomorphism
classes
of
hyperbolic
orbicurves
over
fields
of
cardinality
≤
the
cardinality
of
Q
p
.
(i)
Let
F
be
the
set
of
isomorphism
classes
of
generalized
sub-p-adic
fields
[i.e.,
subfields
of
finitely
generated
extensions
of
the
quotient
field
of
the
ring
of
Witt
vectors
with
coefficients
in
an
algebraic
closure
of
F
p
—
cf.
[Mzk5],
Definition
4.11];
D
=
V
×
F
×
S.
Then
let
us
observe
that:
The
hypotheses
of
Theorem
4.7,
(i),
(ii),
are
satisfied
relative
to
this
D.
Indeed,
it
is
immediate
that
D
is
chain-full;
the
rel-isom-DGC
follows
from
[Mzk5],
Theorem
4.12;
the
prime
p
clearly
serves
as
a
prime
“l”
as
in
the
statement
of
Theorem
4.7.
Moreover,
we
recall
from
[Mzk5],
Lemma
4.14,
that
the
absolute
Galois
group
of
a
generalized
sub-p-adic
field
is
always
slim.
(ii)
Let
F
be
the
set
of
isomorphism
classes
of
sub-p-adic
fields
[i.e.,
subfields
of
finitely
generated
extensions
of
Q
p
—
cf.
[Mzk3],
Definition
15.4,
(i)];
D
=
V×F×S.
Then
let
us
observe
that:
The
hypotheses
of
Theorem
4.7,
(iii),
(iv),
are
satisfied
relative
to
this
D.
Indeed,
it
is
immediate
that
D
is
chain-full;
the
rel-hom-DGC
follows
from
[Mzk3],
Theorem
A;
the
prime
p
clearly
serves
as
a
prime
“l”
as
in
the
statement
of
Theorem
4.7.
Moreover,
we
recall
from
[Mzk3],
Lemma
15.8,
that
the
absolute
Galois
group
of
a
sub-p-adic
field
is
always
slim.
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
I
Example
4.9.
59
Iso-poly-hyperbolic
Orbisurfaces.
(i)
Let
k
be
a
field
of
characteristic
zero.
Then
we
recall
from
[Mzk3],
Definition
a2.1,
that
a
smooth
k-scheme
X
is
called
a
hyperbolically
fibred
surface
if
it
admits
the
structure
of
a
family
of
hyperbolic
curves
[cf.
§0]
over
a
hyperbolic
curve
Y
over
k.
If
X
is
a
smooth,
generically
scheme-like,
geometrically
connected
algebraic
stack
over
k,
then
we
shall
say
that
X
is
an
iso-poly-hyperbolic
orbisurface
[cf.
the
term
“poly-hyperbolic”
as
it
is
defined
in
[Mzk4],
Definition
4.6]
if
X
admits
a
finite
étale
covering
which
is
a
hyperbolically
fibred
surface
over
some
finite
extension
of
k.
def
(ii)
Let
p
be
a
prime
number;
S
=
{Primes}
[where
we
regard
Primes
as
the
unique
non-proper
subset
of
Primes];
F
the
set
of
isomorphism
classes
of
sub-p-adic
fields;
V
the
set
of
isomorphism
classes
of
iso-poly-hyperbolic
orbisurfaces
[cf.
(i)]
over
sub-p-adic
fields;
D
=
V
×
F
×
S.
Then
let
us
observe
that:
The
hypotheses
of
Theorem
4.7,
(i),
(ii),
are
satisfied
relative
to
this
D.
Indeed,
it
is
immediate
that
D
is
chain-full;
the
rel-isom-DGC,
as
well
as
the
slim-
ness
of
the
Δ
i
[for
i
=
1,
2],
follows
immediately
from
[Mzk3],
Theorem
D.
Moreover,
we
recall
from
[Mzk3],
Lemma
15.8,
that
the
absolute
Galois
group
of
a
sub-p-adic
field
is
always
slim.
(iii)
Let
k
be
a
sub-p-adic
field;
X
the
moduli
stack
of
hyperbolic
curves
of
type
(0,
5)
[i.e.,
the
moduli
stack
of
smooth
curves
of
genus
0
with
5
distinct,
unordered
→
X
a
“universal”
pro-finite
étale
covering
of
X;
k
the
algebraic
points]
over
k;
X
→
X.
Then
one
verifies
immediately
that
X
is
an
closure
of
k
determined
by
X
iso-poly-hyperbolic
orbisurface
over
k.
Write
1
→
Δ
→
Π
→
G
→
1
for
the
GSAFG-
extension
defined
by
the
natural
surjection
π
1
(X)
=
Gal(
X/X)
Gal(k/k)
[which
we
regard
as
equipped
with
the
tautological
scheme-theoretic
envelope
given
by
the
identity]
and
Loc(
X/X)
⊆
Chain
trm
(
X/X)
(respectively,
Loc(Π)
⊆
Chain
trm
(Π))
for
the
subcategory
determined
by
the
terminal
morphisms
(respectively,
homomor-
phisms)
which
are
finite
étale
(respectively,
injective).
Then
it
follows
immediately
from
(ii);
Theorem
4.7,
(i),
that
we
have
an
equivalence
of
categories
∼
Loc(
X/X)
→
Loc(Π)
[cf.
[Mzk7],
Theorem
2.4;
[Mzk9],
Theorem
2.3,
(i)].
Moreover,
the
objects
of
these
categories
“Loc(−)”
determined
by
X,
Π
[i.e.,
by
the
unique
chain
of
length
0]
is
terminal
[cf.
[Mzk2],
Theorem
C]
—
i.e.,
a
“core”
[cf.
the
terminology
of
[Mzk7],
§2;
[Mzk8],
§2].
Finally,
we
observe
that
the
theory
of
the
present
§4
admits
a
“tempered
ver-
sion”,
in
the
case
of
hyperbolic
orbicurves
over
MLF’s.
We
begin
by
recalling
basic
facts
concerning
tempered
fundamental
groups.
Let
k
be
an
MLF
of
residue
char-
acteristic
p;
k
an
algebraic
closure
of
k;
X
a
hyperbolic
orbicurve
over
k.
We
shall
use
a
subscript
k
to
denote
the
result
of
a
base-change
from
k
to
k.
Write
π
1
tp
(X);
π
1
tp
(X
k
)
60
SHINICHI
MOCHIZUKI
for
the
tempered
fundamental
groups
of
X,
X
k
[cf.
[André],
§4;
[Mzk10],
Exam-
ples
3.10,
5.6].
Thus,
the
profinite
completion
of
π
1
tp
(X)
(respectively,
π
1
tp
(X
k
))
is
naturally
isomorphic
to
the
usual
étale
fundamental
group
π
1
(X)
(respectively,
π
1
(X
k
)).
If
H
⊆
π
1
tp
(X
k
)
is
an
open
subgroup
of
finite
index,
then
recall
that
the
minimal
co-free
subgroup
of
H
H
co-fr
⊆
H
[cf.
§0]
is
precisely
the
subgroup
of
H
with
the
property
that
the
quotient
H
H/H
co-fr
corresponds
to
the
tempered
covering
of
X
k
determined
by
the
universal
covering
of
the
dual
graph
of
the
special
fiber
of
a
stable
model
of
X
k
—
cf.
[André],
proof
of
Lemma
6.1.1.
Proposition
4.10.
(Basic
Properties
of
Tempered
Fundamental
Groups)
In
the
notation
of
the
above
discussion,
suppose
further
that
φ
:
X
→
Y
is
a
mor-
phism
of
hyperbolic
orbicurves
over
k.
For
Z
=
X,
Y
,
let
us
write
def
tp
Π
tp
Z
=
π
1
(Z);
def
tp
Δ
tp
Z
=
π
1
(Z
k
)
tp
tp
tp
and
denote
the
profinite
completions
of
Π
tp
Z
,
Δ
Z
by
Π
Z
,
Δ
Z
,
respectively;
in
the
following,
all
“co-free
completions”
[cf.
§0]
of
open
subgroups
of
finite
index
tp
in
Π
tp
X
(respectively,
Δ
X
)
will
be
with
respect
to
[the
intersections
of
such
open
tp
tp
tp
subgroups
with]
the
subgroup
Δ
tp
X
⊆
Π
X
(respectively,
Δ
X
⊆
Δ
X
).
Then:
tp
tp
∼
(i)
The
natural
homomorphism
Π
tp
X
→
Π
X
→
π
1
(X)
(respectively,
Δ
X
→
∼
tp
→
π
1
(X
))
is
injective.
In
fact,
if
H
⊆
Δ
tp
is
any
characteristic
open
Δ
X
X
k
co-fr
tp
co-fr
subgroup
of
finite
index,
then
Π
tp
/H
,
Δ
/H
inject
into
their
respective
X
tp
profinite
completions.
In
particular,
π
1
(X)
(respectively,
π
1
tp
(X
k
))
is
naturally
isomorphic
to
its
π
1
(X)-co-free
completion
(respectively,
π
1
(X
k
)-co-free
com-
pletion)
[cf.
§0].
tp
tp
tp
(ii)
Π
tp
X
(respectively,
Δ
X
)
is
normally
terminal
in
Π
X
(respectively,
Δ
X
).
(iii)
Suppose
that
φ
is
either
a
de-cuspidalization
morphism
[i.e.,
an
open
immersion
whose
image
is
the
complement
of
a
single
k-valued
point
of
Y
—
cf.
Definition
4.2,
(i),
(c)]
or
a
de-orbification
morphism
[i.e.,
a
partial
coarsification
morphism
which
is
an
isomorphism
over
the
complement
of
a
single
k-valued
point
tp
of
Y
—
cf.
Definition
4.2,
(i),
(d)].
Then
the
natural
homomorphism
Π
tp
X
→
Π
Y
tp
(respectively,
Δ
tp
X
→
Δ
Y
)
may
be
reconstructed
—
“group-theoretically”
—
tp
(respectively,
Δ
tp
Δ
tp
)
as
the
natural
tp
Π
from
its
profinite
completion
Π
X
Y
X
Y
tp
morphism
from
Π
tp
(respectively,
Δ
)
to
the
co-free
completion
of
Π
tp
X
X
X
with
tp
tp
)
[cf.
§0].
(respectively,
Δ
respect
to
Π
Y
Y
(iv)
Let
l
∈
Primes.
If
J
⊆
Δ
tp
X
is
an
open
subgroup
of
finite
index,
write
[l]
J
→
J
for
the
co-free
completion
of
J
with
respect
to
the
maximal
pro-l
quotient
of
the
profinite
completion
of
J.
Let
H
⊆
Δ
tp
X
be
an
open
subgroup
of
finite
index.
Suppose
that
l
=
p.
Then
the
dual
graph
Γ
H
of
the
special
fiber
of
a
stable
model
of
the
covering
of
X
k
corresponding
to
H
determines
verticial
and
edge-like
subgroups
of
H
[l]
[i.e.,
decomposition
groups
of
the
vertices
and
edges
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
I
61
of
Γ
H
—
cf.
[Mzk10],
Theorem
3.7,
(i),
(iii)].
The
verticial
(respectively,
edge-
like)
subgroups
of
H
[l]
may
be
characterized
—
“group-theoretically”
—
as
the
maximal
compact
subgroups
(respectively,
nontrivial
intersections
of
two
distinct
maximal
compact
subgroups)
of
H
[l]
.
In
particular,
the
graph
Γ
H
may
be
reconstructed
—
“group-theoretically”
—
from
the
verticial
and
edge-
like
subgroups
of
H
[l]
,
together
with
their
various
mutual
inclusion
relations.
(v)
The
prime
number
p
may
be
characterized
—
“group-theoretically”
—
as
the
unique
prime
number
l
such
that
their
exist
open
subgroups
H,
J
⊆
Δ
tp
X
of
finite
index,
together
with
distinct
prime
numbers
l
1
,
l
2
,
satisfying
the
following
properties:
(a)
H
is
a
normal
subgroup
of
J
of
index
l;
(b)
for
i
=
1,
2,
the
outer
action
of
J
on
H
[l
i
]
[cf.
(iv)]
fixes
[the
conjugacy
class
in
H
[l
i
]
of
]
and
induces
the
trivial
outer
action
on
some
maximal
compact
subgroup
of
H
[l
i
]
[cf.
(iv)].
(vi)
Let
l
be
a
prime
number
=
p
[a
“group-theoretic”
condition,
by
(v)!];
H
⊆
Δ
tp
X
an
open
subgroup
of
finite
index.
Then
the
set
of
cusps
of
the
covering
of
X
k
corresponding
to
H
may
be
characterized
—
“group-theoretically”
—
as
the
set
of
conjugacy
classes
in
H
[l]
of
the
commensurators
in
H
[l]
of
the
images
in
H
[l]
of
edge-like
subgroups
of
J
[l]
[cf.
(iv)],
where
J
⊆
H
is
an
open
subgroup
of
finite
index,
which
are
not
contained
in
edge-like
subgroups
of
H
[l]
.
In
particular,
by
allowing
H
to
vary,
this
yields
a
[“group-theoretic”]
characterization
of
the
tp
decomposition
groups
of
cusps
in
Δ
tp
X
,
Π
X
[i.e.,
a
“tempered
version”
of
Lemma
4.5,
(v)].
Proof.
Assertion
(i)
follows
immediately
from
the
discussion
at
the
beginning
of
[Mzk10],
§6
[cf.
also
the
discussion
of
[André],
§4.5].
Assertion
(ii)
is
the
content
of
[Mzk10],
Lemma
6.1,
(ii),
(iii)
[cf.
also
[André],
Corollary
6.2.2].
Assertion
(iii)
follows
immediately
from
assertion
(i).
Assertion
(iv)
follows
immediately
from
[Mzk10],
Theorem
3.7,
(iv);
[Mzk10],
Corollary
3.9
[cf.
also
the
proof
of
[Mzk10],
Corollary
3.11].
Assertions
(v),
(vi)
amount
to
summaries
of
the
relevant
portions
of
the
proof
of
[Mzk10],
Corollary
3.11.
Here,
in
assertion
(v),
we
observe
that
at
least
one
of
the
l
i
is
=
p;
thus,
for
this
choice
of
l
i
,
the
action
of
J
fixes
and
induces
the
trivial
outer
action
on
some
verticial
subgroup
of
H
[l
i
]
.
Remark
4.10.1.
It
is
not
clear
to
the
author
at
the
time
of
writing
how
to
prove
a
version
of
Proposition
4.10,
(vi),
for
decomposition
groups
of
closed
points
which
are
not
cusps
[i.e.,
a
“tempered
version”
of
Lemma
4.1,
(iii)].
Remark
4.10.2.
A
certain
fact
applied
in
the
portion
of
the
proof
of
[Mzk10],
Corollary
3.11,
summarized
in
Proposition
4.10,
(vi),
is
only
given
a
somewhat
sketchy
proof
in
loc.
cit.
A
more
detailed
treatment
of
this
fact
is
given
in
[Mzk15],
Corollary
2.11.
Now
we
are
ready
to
state
the
tempered
version
of
Definition
4.2.
Definition
4.11.
In
the
notation
of
the
above
discussion,
let
1
→
Δ
→
Π
→
G
→
1
62
SHINICHI
MOCHIZUKI
be
an
extension
of
topological
groups
that
is
isomorphic
to
the
natural
extension
∼
1
→
π
1
tp
(X
k
)
→
π
1
tp
(X)
→
Gal(k/k)
→
1
via
some
isomorphism
α
:
π
1
tp
(X)
→
Π,
for
the
profinite
which
we
shall
refer
to
as
a
scheme-theoretic
envelope.
Write
Π
→
X
for
the
pro-finite
étale
covering
of
X
determined
by
completion
of
Π;
X
=
Gal(
X/X)];
the
completion
of
α
[so
Π
k
for
the
resulting
field
extension
of
k.
In
a
similar
vein,
we
shall
write
Π
for
the
projective
system
of
topological
groups
determined
by
the
open
subgroups
of
finite
index
of
Π
[cf.
Definition
4.2].
Then:
(i)
We
shall
refer
to
as
an
[Π-]chain
[of
length
n]
[where
n
≥
0
is
an
integer]
any
finite
sequence
Π
0
Π
1
.
.
.
Π
n−1
Π
n
j
,
each
of
topological
groups
Π
j
[for
j
=
0,
.
.
.
,
n]
with
slim
profinite
completions
Π
→
Π
j
which
is
of
DOF-type
equipped
with
a
“rigidifying
homomorphism”
ρ
j
:
Π
onto
a
dense
subgroup
of
[i.e.,
which
maps
some
member
of
the
projective
system
Π
an
open
subgroup
of
finite
index
of
Π
j
—
cf.
§0]
satisfying
the
following
conditions:
→
Π].
(0
tp
)
Π
0
=
Π
[equipped
with
its
natural
rigidifying
homomorphism
Π
(1
tp
)
There
exists
a
[uniquely
determined]
surjection
Π
j
G
j
,
where
G
j
⊆
G
is
an
open
subgroup,
that
is
compatible
with
ρ
j
and
the
natural
composite
→
Π
G.
morphism
Π
(2
tp
)
Each
kernel
def
Δ
j
=
Ker(Π
j
G
j
→
G)
j
.
has
a
slim,
nontrivial
profinite
completion
Δ
(3
tp
)
The
topological
groups
Π
j
,
Δ
j
are
residually
finite.
Also,
we
shall
refer
j
any
Δ
j
-conjugate
of
the
com-
to
as
a
cuspidal
decomposition
group
in
Δ
j
of
a
nontrivial
image
via
ρ
j
of
the
inverse
image
in
Π
of
mensurator
in
Δ
the
decomposition
group
in
Δ
[determined
by
α]
of
a
cusp
of
X.
(4
tp
)
Each
“Π
j
Π
j+1
”
[for
j
=
0,
.
.
.
,
n
−
1]
is
an
“elementary
operation”,
as
defined
below.
Here,
an
elementary
operation
“Π
j
Π
j+1
”
is
defined
to
consist
of
the
datum
of
an
“operation
homomorphism”
φ
of
DOF-type
either
from
Π
j
to
Π
j+1
or
from
Π
j+1
to
Π
j
which
is
compatible
with
ρ
j
,
ρ
j+1
,
and,
moreover,
is
of
one
of
the
following
four
types:
(a)
Type
:
In
this
case,
the
elementary
operation
Π
j
Π
j+1
consists
of
an
immersion
of
OF-type
[cf.
§0]
φ
:
Π
j+1
→
Π
j
.
(b)
Type
:
In
this
case,
the
elementary
operation
Π
j
Π
j+1
consists
of
an
immersion
of
OF-type
[cf.
§0]
φ
:
Π
j
→
Π
j+1
.
(c)
Type
•:
In
this
case,
the
elementary
operation
Π
j
Π
j+1
consists
of
a
dense
homomorphism
φ
:
Π
j
→
Π
j+1
which
is
isomorphic
to
the
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
I
63
co-free
completion
of
Π
j
with
respect
to
the
induced
profinite
quotient
is
topologically
j+1
[and
the
subgroup
Δ
j
],
such
that
Ker(
φ)
j
Π
φ
:
Π
j
such
that
normally
generated
by
a
cuspidal
decomposition
group
C
in
Δ
j
.
C
is
contained
in
some
normal
open
torsion-free
subgroup
of
Δ
(d)
Type
:
In
this
case,
the
elementary
operation
Π
j
Π
j+1
consists
of
a
dense
homomorphism
φ
:
Π
j
→
Π
j+1
which
is
isomorphic
to
the
co-free
completion
of
Π
j
with
respect
to
the
induced
profinite
quotient
is
topologically
j+1
[and
the
subgroup
Δ
j
],
such
that
Ker(
φ)
j
Π
φ
:
Π
j
.
normally
generated
by
a
finite
closed
subgroup
of
Δ
Thus,
any
Π-chain
determines
a
sequence
of
symbols
∈
{,
,
•,
}
[corresponding
to
the
types
of
elementary
operations
in
the
Π-chain],
which
we
shall
refer
to
as
the
type-chain
associated
to
the
Π-chain.
(ii)
An
isomorphism
between
two
Π-chains
with
identical
type-chains
[hence
of
the
same
length]
∼
(Π
0
.
.
.
Π
n
)
→
(Ψ
0
.
.
.
Ψ
n
)
∼
is
defined
to
be
a
collection
of
isomorphisms
of
topological
groups
Π
j
→
Ψ
j
[for
j
=
0,
.
.
.
,
n]
that
are
compatible
with
the
rigidifying
homomorphisms.
[Here,
we
note
that
the
condition
of
compatibility
with
the
rigidifying
homomorphisms
implies
[since
all
of
the
topological
groups
involved
are
residually
finite
with
slim
profinite
completions]
that
every
automorphism
of
a
Π-chain
is
given
by
the
identity,
and
that
every
isomorphism
of
Π-chains
of
the
same
length
is
compatible
with
the
respective
operation
homomorphisms.]
Thus,
one
obtains
a
category
Chain(Π)
whose
objects
are
the
Π-chains
[with
arbitrary
associated
type-chain],
and
whose
morphisms
are
the
isomorphisms
between
Π-chains
[with
identical
type-chains].
A
terminal
homomorphism
between
two
Π-chains
[with
arbitrary
associated
type-
chains]
(Π
0
.
.
.
Π
n
)
→
(Ψ
0
.
.
.
Ψ
m
)
is
defined
to
be
an
outer
homomorphism
of
DOF-type
[cf.
§0;
[Mzk10],
Theorem
6.4]
Π
n
→
Ψ
m
that
is
compatible
[up
to
composition
with
an
inner
automorphism]
with
the
open
homomorphisms
Π
n
→
G,
Ψ
m
→
G.
Thus,
one
obtains
a
category
Chain
trm
(Π)
whose
objects
are
the
Π-chains
[with
arbitrary
associated
type-chain],
and
whose
morphisms
are
the
terminal
homomorphisms
between
Π-chains;
write
Chain
iso-trm
(Π)
⊆
Chain
trm
(Π)
for
the
subcategory
determined
by
the
terminal
isomorphisms
[i.e.,
the
isomor-
phisms
of
Chain
trm
(Π)].
Thus,
it
follows
immediately
from
the
definitions
that
we
64
SHINICHI
MOCHIZUKI
obtain
natural
functors
Chain(Π)
→
Chain
iso-trm
(Π)
→
Chain
trm
(Π).
Finally,
we
have
(sub)categories
Chain
iso-trm
(Π){−}
⊆
Chain
iso-trm
(Π);
def
DLoc(Π)
=
Chain
trm
(Π){,
•};
Chain
trm
(Π){−}
⊆
Chain
trm
(Π)
def
ÉtLoc(Π)
=
Chain
iso-trm
(Π){,
}
[cf.
Definition
4.2,
(v)].
Remark
4.11.1.
Just
as
in
the
profinite
case
[i.e.,
Remark
4.2.1],
we
have
natural
functors
Chain(
X/X)
→
Chain(Π)
→
Chain(
Π)
Chain
iso-trm
(
X/X)
→
Chain
iso-trm
(Π)
→
Chain
iso-trm
(
Π)
Chain
trm
(
X/X)
→
Chain
trm
(Π)
→
Chain
trm
(
Π)
—
where
we
apply
Proposition
4.10,
(iii),
in
the
construction
of
the
first
arrow
in
each
line;
the
second
arrow
in
each
line
is
the
natural
functor
obtained
by
profinite
completion;
the
various
composite
functors
of
the
two
functors
in
each
line
are
the
natural
functors
of
Remark
4.2.1.
Remark
4.11.2.
pered
case.
A
similar
remark
to
Remark
4.2.2
applies
in
the
present
tem-
Theorem
4.12.
(Tempered
Chains
of
Elementary
Operations)
For
i
=
1,
2,
let
k
i
be
an
MLF
of
residue
characteristic
p
i
;
k
i
an
algebraic
closure
of
k
i
;
X
i
a
hyperbolic
orbicurve
over
k
i
;
1
→
Δ
i
→
Π
i
→
G
i
→
1
an
extension
of
topological
groups
that
is
isomorphic
to
the
natural
exten-
sion
1
→
π
1
tp
((X
i
)
k
i
)
→
π
1
tp
(X
i
)
→
Gal(k
i
/k
i
)
→
1
via
some
scheme-theoretic
∼
envelope
α
i
:
π
1
tp
(X
i
)
→
Π
i
.
Let
∼
φ
:
Π
1
→
Π
2
be
an
isomorphism
of
topological
groups.
Then:
(i)
The
natural
functors
[cf.
Remark
4.11.1]
i
/X
i
)
→
Chain(Π
i
);
Chain(
X
i
/X
i
)
→
Chain
iso-trm
(Π
i
)
Chain
iso-trm
(
X
i
/X
i
)
→
ÉtLoc(Π
i
)
ÉtLoc(
X
i
/X
i
)
→
Chain
trm
(Π
i
);
Chain
trm
(
X
i
/X
i
)
→
DLoc(Π
i
)
DLoc(
X
are
equivalences
of
categories
that
are
compatible
with
passing
to
type-chains.
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
I
65
∼
(ii)
We
have
p
1
=
p
2
;
the
isomorphism
φ
induces
isomorphisms
φ
Δ
:
Δ
1
→
Δ
2
,
∼
φ
G
:
G
1
→
G
2
,
as
well
as
equivalences
of
categories
∼
Chain(Π
1
)
→
Chain(Π
2
);
∼
Chain
iso-trm
(Π
1
)
→
Chain
iso-trm
(Π
2
)
∼
ÉtLoc(Π
1
)
→
ÉtLoc(Π
2
)
∼
Chain
trm
(Π
1
)
→
Chain
trm
(Π
2
);
∼
DLoc(Π
1
)
→
DLoc(Π
2
)
that
are
compatible
with
passing
to
type-chains
and
functorial
in
φ.
Proof.
In
light
of
Proposition
4.10,
(iii),
together
with
the
“tempered
anabelian
theorem”
of
[Mzk10],
Theorem
6.4,
the
proof
of
Theorem
4.12
is
entirely
similar
to
the
proof
of
Theorem
4.7.
[Here,
we
note
that
in
the
case
of
de-cuspidalization
operations,
instead
of
applying
the
de-cuspidalization
portion
of
Proposition
4.10,
(iii),
one
may
instead
apply
the
“group-theoretic”
characterization
of
Proposition
4.10,
(vi).]
Also,
we
recall
that
the
portion
of
assertion
(ii)
concerning,
“p
1
=
p
2
”,
“φ
Δ
”,
“φ
G
”
follows
immediately
[by
considering
the
profinite
completion
of
φ]
from
Theorem
2.14,
(i).
Remark
4.12.1.
A
similar
remark
to
Remark
4.7.1
applies
in
the
present
tem-
pered
case
[cf.
[Mzk10],
Theorem
6.8].
66
SHINICHI
MOCHIZUKI
Appendix:
The
Theory
of
Albanese
Varieties
In
the
present
Appendix,
we
review
the
basic
theory
of
Albanese
varieties
[cf.,
e.g.,
[NS],
[Serre1],
[Chev],
[BS],
[SS]],
as
it
will
be
applied
in
the
present
paper.
One
of
our
aims
here
is
to
present
the
theory
in
modern
scheme-theoretic
language
[i.e.,
as
opposed
to
[NS],
[Serre1],
[Chev]],
but
without
resorting
to
the
introduction
of
motives
and
derived
categories,
as
in
[BS],
[SS].
Put
another
way,
although
there
is
no
doubt
that
the
content
of
the
present
Appendix
is
implicit
in
the
literature,
the
lack
of
an
appropriate
reference
that
discusses
this
material
explicitly
seemed
to
the
author
to
constitute
sufficient
justification
for
the
inclusion
of
a
detailed
discussion
of
this
material
in
the
present
paper.
In
the
following
discussion,
we
fix
a
perfect
field
k,
together
with
an
algebraic
closure
k
of
k.
The
result
of
base-change
[of
k-schemes
and
morphisms
of
k-schemes]
def
from
k
to
k
will
be
denoted
by
means
of
a
subscript
“k”.
Write
G
k
=
Gal(k/k)
for
the
absolute
Galois
group
of
k.
We
will
apply
basic
well-known
properties
of
commutative
group
schemes
of
finite
type
over
k
without
further
explanation.
In
particular,
we
recall
the
following:
(I)
The
category
of
such
group
schemes
is
abelian
[cf.,
e.g.,
[SGA3-1],
VI
A
,
5.4];
subgroup
schemes
are
always
closed
[cf.,
e.g.,
[SGA3-1],
VI
B
,
1.4.2];
reduced
group
schemes
over
k
are
k-smooth
[cf.,
e.g.,
[SGA3-1],
VI
A
,
1.3.1].
(II)
Every
connected
reduced
subquotient
of
a
semi-abelian
variety
over
k
[i.e.,
an
extension
of
an
abelian
variety
by
a
torus]
is
itself
a
semi-abelian
variety
over
k.
[Indeed,
this
may
be
verified
easily
by
applying
a
well-
known
theorem
of
Chevalley
[cf.,
e.g.,
[Con],
for
a
treatment
of
this
result
in
modern
language;
[Bor],
Theorems
10.6,
10.9],
to
the
effect
that
any
smooth
connected
commutative
group
scheme
over
k
may
be
written
as
an
extension
of
a
semi-abelian
variety
by
a
successive
extension
of
copies
of
the
additive
group
(G
a
)
k
.]
(III)
Let
φ
:
B
→
A
be
a
connected
finite
étale
Galois
covering
of
a
semi-
abelian
variety
A
over
k,
with
identity
element
0
A
∈
A(k),
such
that
(φ
−1
(0
A
))(k)
=
∅,
and
the
degree
of
φ
is
prime
to
the
characteristic
of
k.
Then
each
element
of
b
∈
(φ
−1
(0
A
))(k)
determines
on
B
a
unique
structure
of
semi-abelian
variety
over
k
on
B
such
that
b
is
the
identity
element
of
the
group
B(k),
and
φ
is
a
homomorphism
of
group
schemes
over
k.
[Indeed
this
may
be
verified
easily
by
applying
the
theorem
of
Chevalley
quoted
in
(II)
above.]
Note,
moreover,
that
in
this
situation,
if
k
=
k,
then
we
obtain
an
inclusion
Gal(B/A)
→
B(k),
which
implies,
in
particular,
that
the
covering
φ
is
abelian,
and,
moreover,
appears
as
a
subcovering
of
a
covering
A
→
A
given
by
multiplication
by
some
n
invertible
in
k.
Definition
A.1.
(i)
A
variety
over
k,
or
k-variety,
is
defined
to
be
a
geometrically
integral
separated
scheme
of
finite
type
over
k.
A
k-variety
will
be
called
complete
if
it
is
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
I
67
proper
over
k.
We
shall
refer
to
a
pair
(V,
v),
where
V
is
a
k-variety
and
v
∈
V
(k),
as
a
pointed
variety
over
k;
a
morphism
of
pointed
varieties
over
k,
or
pointed
k-
morphism,
(V,
v)
→
(W,
w)
[which
we
shall
often
simply
write
V
→
W
,
when
v,
w
are
fixed]
is
a
morphism
of
k-varieties
that
maps
v
→
w.
Any
reduced
group
scheme
G
over
k
has
a
natural
structure
of
pointed
variety
over
k
determined
by
the
identity
element
0
G
∈
G(k).
If
G,
H
are
group
schemes
over
k,
then
we
shall
refer
to
a
k-morphism
G
→
H
as
a
[k-]trans-homomorphism
if
it
factors
as
the
composite
of
a
homomorphism
of
group
schemes
G
→
H
over
k
with
an
automorphism
of
H
given
by
translation
by
an
element
of
H(k).
If
V
is
a
k-variety,
then
we
shall
use
the
notation
π
1
(V
)
to
denote
the
étale
fundamental
group
[relative
to
an
appropriate
choice
of
basepoint]
of
V
.
Thus,
we
have
a
natural
exact
sequence
of
fundamental
groups
1
→
π
1
(V
k
)
→
π
1
(V
)
→
G
k
→
1.
Let
Σ
k
⊆
Primes
[cf.
§0]
be
the
set
of
primes
invertible
in
k;
use
the
superscript
“(Σ
k
)”
to
denote
the
maximal
pro-Σ
k
quotient
of
a
profinite
group;
if
V
is
a
k-variety,
then
we
shall
write
def
Δ
V
=
π
1
(V
k
)
(Σ
k
)
;
def
Π
V
=
π
1
(V
)/Ker(π
1
(V
k
)
π
1
(V
k
)
(Σ
k
)
)
for
the
resulting
geometrically
pro-Σ
k
fundamental
groups,
so
we
have
a
natural
exact
sequence
of
fundamental
groups
1
→
Δ
V
→
Π
V
→
G
k
→
1.
(ii)
Let
C
be
a
class
of
commutative
group
schemes
of
finite
type
over
k.
If
A
is
a
group
scheme
over
k
that
belongs
to
the
class
C,
then
we
shall
write
A
∈
C.
If
(V,
v)
is
a
pointed
k-variety,
then
we
shall
refer
to
a
morphism
of
pointed
k-varieties
φ
:
V
→
A
as
a
C-Albanese
morphism
if
A
∈
C
[so
A
is
equipped
with
a
point
0
A
∈
A(k),
as
discussed
in
(i)],
and,
moreover,
for
any
pointed
k-morphism
φ
:
V
→
A
,
where
A
∈
C,
there
exists
a
unique
homomorphism
ψ
:
A
→
A
of
group
schemes
over
k
such
that
φ
=
ψ
◦
φ.
In
this
situation,
A
will
also
be
referred
to
as
the
C-Albanese
variety
of
V
.
We
shall
write
C
k
ab
for
the
class
of
abelian
varieties
over
k
and
C
k
s-ab
for
the
class
of
semi-abelian
varieties
over
k.
When
C
=
C
k
s-ab
,
the
term
“C-Albanese”,
will
often
be
abbreviated
“Albanese”.
(iii)
If
X
is
a
k-variety
(respectively,
noetherian
scheme)
which
admits
a
log
structure
such
that
the
resulting
log
scheme
X
log
is
log
smooth
over
k
[where
we
regard
Spec(k)
as
equipped
with
the
trivial
log
structure]
(respectively,
log
regular
[cf.
[Kato]]),
then
we
shall
refer
to
X
as
k-toric
(respectively,
absolutely
toric)
and
to
X
log
as
a
torifier,
or
torifying
log
scheme,
for
X.
[Thus,
“k-toric”
implies
“absolutely
toric”.]
(iv)
If
k
is
of
positive
characteristic,
then,
for
any
k-scheme
X
and
integer
n
n
≥
1,
we
shall
write
X
F
for
the
result
of
base-changing
X
by
the
n-th
iterate
of
the
Frobenius
morphism
on
k;
thus,
we
obtain
a
k-linear
relative
Frobenius
n
n
def
morphism
Φ
nX
:
X
→
X
F
.
If
k
is
of
characteristic
zero,
then
we
set
X
F
=
X,
def
Φ
nX
=
id
X
,
for
integers
n
≥
1.
If
φ
:
X
→
Y
is
a
morphism
of
k-schemes,
then
we
shall
refer
to
φ
as
a
sub-Frobenius
morphism
if,
for
some
integer
n
≥
1,
there
n
n
exists
a
k-morphism
ψ
:
Y
→
X
F
such
that
ψ
◦
φ
=
Φ
nX
,
φ
F
◦
ψ
=
Φ
nY
.
[Thus,
in
characteristic
zero,
a
sub-Frobenius
morphism
is
simply
an
automorphism.]
68
SHINICHI
MOCHIZUKI
Remark
A.1.1.
As
is
well-known,
if
V
is
a
k-variety,
then
Φ
nV
induces
an
∼
isomorphism
Π
V
→
Π
V
F
n
,
for
all
integers
n
≥
1.
Note
that
this
implies
that
every
∼
sub-Frobenius
morphism
V
→
W
of
k-varieties
induces
isomorphisms
Π
V
→
Π
W
,
∼
Δ
V
→
Δ
W
.
Before
proceeding,
we
review
the
following
well-known
result.
Lemma
A.2.
(Morphisms
to
Abelian
and
Semi-abelian
Schemes)
Let
S
be
a
noetherian
scheme;
X
an
S-scheme
whose
underlying
scheme
is
absolutely
toric;
A
an
abelian
scheme
over
S
(respectively,
a
semi-abelian
scheme
over
S
which
is
an
extension
of
an
abelian
scheme
B
→
S
by
a
torus
T
→
S);
V
⊆
X
an
open
subscheme
whose
complement
in
X
is
of
codimension
≥
1
(respectively,
≥
2)
in
X.
Then
any
morphism
of
S-schemes
V
→
A
extends
uniquely
to
X.
Proof.
First,
we
consider
the
case
where
A
is
an
abelian
scheme.
If
X
is
regular,
then
Lemma
A.2
follows
from
[BLR],
§8.4,
Corollary
6.
When
X
is
an
arbitrary
absolutely
toric
scheme
with
torifier
X
log
,
we
reduce
immediately
to
the
case
where
X
is
strictly
henselian,
hence
admits
a
resolution
of
singularities
[cf.,
e.g.,
[Mzk4],
§2]
Y
log
→
X
log
∼
—
i.e.,
a
log
étale
morphism
of
log
schemes
which
induces
an
isomorphism
U
Y
→
U
X
between
the
respective
interiors
such
that
Y
log
arises
from
a
divisor
with
normal
crossings
in
a
regular
scheme
Y
.
Since
the
“regular
case”
has
already
been
settled,
we
may
assume
that
U
X
⊆
V
;
also,
it
follows
that
the
restriction
U
Y
→
A
to
U
Y
of
the
resulting
morphism
U
X
→
A
extends
uniquely
to
a
morphism
Y
→
A.
def
The
graph
of
this
morphism
determines
a
closed
subscheme
Z
⊆
A
Y
=
A
×
S
Y
.
def
Moreover,
by
considering
the
image
of
Z
under
the
morphism
A
Y
→
A
X
=
A
×
S
X
of
proper
X-schemes,
we
conclude
from
“Zariski’s
main
theorem”
[since
X
is
normal]
that
to
obtain
the
[manifestly
unique,
since
V
is
schematically
dense
in
X]
desired
extension
X
→
A,
it
suffices
to
show
that
the
fibers
of
Y
→
X
map
to
points
of
A.
On
the
other
hand,
as
is
observed
in
the
discussion
of
[Mzk4],
§2,
each
irreducible
component
of
the
fiber
of
Y
→
X
at
a
point
x
∈
X
is
a
rational
variety
over
the
residue
field
k(x)
at
x,
hence
maps
to
a
point
in
the
abelian
variety
def
A
x
=
A
×
S
k(x)
[cf.,
e.g.,
[BLR],
§10.3,
Theorem
1,
(b),
(c)].
This
completes
the
proof
of
Lemma
A.2
in
the
non-resp’d
case.
Thus,
to
complete
the
proof
of
Lemma
A.2
in
the
resp’d
case,
we
may
assume
that
A
=
T
is
a
torus
over
S.
In
fact,
by
étale
descent,
we
may
even
assume
that
T
is
a
split
torus
over
S.
Then
it
suffices
to
show
that
if
L
is
any
line
bundle
on
X
that
admits
a
generating
section
s
V
∈
Γ(V,
L),
then
it
follows
that
s
V
extends
to
a
generating
section
of
L
over
X.
But
since
X
is
normal,
this
follows
immediately
from
[SGA2],
XI,
3.4;
[SGA2],
XI,
3.11.
Proposition
A.3.
(Basic
Properties
of
Albanese
Varieties)
Let
C
∈
ab
s-ab
{C
k
,
C
k
};
φ
V
:
V
→
A,
φ
W
:
W
→
B
C-Albanese
morphisms.
Then:
(i)
(Base-change)
Let
k
be
an
algebraic
field
extension
of
k;
denote
the
result
of
base-change
[of
k-schemes
and
morphisms
of
k-schemes]
from
k
to
k
by
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
I
69
means
of
a
subscript
“k
”.
If
C
=
C
k
ab
(respectively,
C
=
C
k
s-ab
),
then
set
C
=
C
k
ab
(respectively,
C
=
C
k
s-ab
).
Then
(φ
V
)
k
:
V
k
→
A
k
is
a
C
-Albanese
morphism.
(ii)
(Functoriality)
Given
any
k-morphism
β
V
:
V
→
W
,
there
exists
a
unique
k-trans-homomorphism
β
A
:
A
→
B
such
that
φ
W
◦
β
V
=
β
A
◦
φ
V
.
If,
moreover,
β
V
is
pointed,
then
β
A
is
a
homomorphism.
n
n
F
→
(iii)
(Relative
Frobenius
Morphisms)
For
any
integer
n
≥
1,
φ
F
V
:
V
n
n
A
F
is
a
C-Albanese
morphism.
If,
moreover,
in
(ii),
φ
W
=
φ
F
,
β
=
Φ
V
V
V
,
n
then
β
A
=
Φ
A
.
n
(iv)
(Sub-Frobenius
Morphisms)
If,
in
(ii),
β
V
is
a
sub-Frobenius
mor-
phism,
then
so
is
β
A
.
(v)
(Toric
Open
Immersions)
Suppose,
in
(ii),
that
β
V
is
an
open
im-
mersion,
that
W
is
k-toric,
and
that
if
C
=
C
k
ab
(respectively,
C
=
C
k
s-ab
),
then
the
codimension
of
the
complement
of
the
image
of
β
V
in
W
is
≥
1
(respectively,
≥
2).
Then
β
A
is
an
isomorphism.
(vi)
(Dominant
Quotients)
If,
in
(ii),
β
V
is
dominant,
then
β
A
is
surjec-
tive.
(vii)
(Surjectivity
of
Fundamental
Groups)
The
[outer]
homomorphisms
Π
φ
V
:
Π
V
→
Π
A
,
Δ
φ
V
:
Δ
V
→
Δ
A
induced
by
φ
V
are
surjective.
(viii)
(Semi-abelian
versus
Abelian
Albanese
Morphisms)
Suppose
that
C
=
C
k
s-ab
.
Write
A
A
ab
for
the
maximal
quotient
of
group
schemes
over
k
such
that
A
ab
∈
C
k
ab
.
Then
the
composite
morphism
V
→
A
A
ab
is
a
C
k
ab
-Albanese
morphism.
(ix)
(Group
Law
Generation)
For
integers
n
≥
1,
write
ζ
n
:
V
×
k
.
.
.
×
k
V
→
A
(v
1
,
.
.
.
,
v
n
)
→
n
j=1
v
j
for
the
morphism
from
the
product
over
k
of
n
copies
of
V
to
A
given
by
adding
the
images
under
φ
V
of
the
points
in
the
n
factors.
Then
there
exists
an
integer
N
such
that
ζ
n
is
surjective
for
all
n
≥
N
.
In
particular,
if
V
is
proper,
then
so
is
A.
Proof.
To
verify
assertion
(i),
we
may
assume
that
k
is
a
finite
[hence
necessarily
étale,
since
k
is
perfect]
extension
of
k.
Then
assertion
(i)
follows
immediately
by
considering
the
Weil
restriction
functor
W
k
/k
(−)
from
k
to
k.
That
is
to
say,
it
is
immediate
that
W
k
/k
(−)
takes
objects
in
C
to
objects
in
C.
Thus,
to
give
a
k
-morphism
V
k
→
A
(respectively,
A
k
→
A
)
is
equivalent
to
giving
a
k-morphism
V
→
W
k
/k
(A
)
(respectively,
A
→
W
k
/k
(A
)).
This
completes
the
proof
of
assertion
(i).
Assertions
(ii),
(iii)
follow
immediately
from
the
definition
of
a
“C-Albanese
morphism”;
assertion
(iv)
follows
immediately
from
assertion
(iii).
Assertion
(v)
follows
immediately
from
the
definition
of
a
“C-Albanese
morphism”,
in
light
of
Lemma
A.2.
70
SHINICHI
MOCHIZUKI
Assertion
(vi)
follows
from
the
definition
of
a
“C-Albanese
morphism”,
by
ar-
guing
as
follows:
First,
we
observe
that
β
V
is
an
epimorphism
in
the
category
of
schemes.
Also,
we
may
assume
without
loss
of
generality
that
β
V
is
pointed.
Now
consider
the
composite
β
◦
φ
W
:
W
→
B/C
of
φ
W
:
W
→
B
with
the
natural
quo-
def
tient
morphism
β
:
B
B/C,
where
we
write
C
=
Im(β
A
)
⊆
B
[so
C
∈
C].
Since
β
◦
φ
W
has
the
same
restriction
[via
β
V
]
to
V
as
the
constant
pointed
morphism
W
→
B/C,
we
thus
conclude
that
β
◦
φ
W
is
constant,
i.e.,
that
Im(φ
W
)
⊆
C.
But,
by
the
definition
of
a
“C-Albanese
morphism”,
this
implies
the
existence
of
a
section
B
→
C
of
the
natural
inclusion
C
→
B
[i.e.,
such
that
the
composite
B
→
C
→
B
is
equal
to
the
identity],
hence
that
B
=
C,
as
desired.
In
a
similar
vein,
assertion
(vii)
follows
from
the
definition
of
a
“C-Albanese
morphism”,
by
ob-
serving
that
if
Π
φ
V
:
Π
V
→
Π
A
fails
to
surject,
then
[after
possibly
replacing
k
by
a
finite
extension
of
k,
which
is
possible,
by
assertion
(i)]
it
follows
that
φ
V
:
V
→
A
factors
V
→
C
→
A,
where
the
morphism
C
→
A
is
a
nontrivial
finite
étale
Galois
covering,
with
C
geometrically
connected
over
k,
so
C
∈
C.
But
this
implies,
by
the
definition
of
a
“C-Albanese
morphism”,
the
existence
of
a
section
A
→
C
of
the
surjection
C
A
[i.e.,
such
that
the
composite
A
→
C
A
is
the
identity],
hence
∼
that
this
surjection
is
an
isomorphism
C
→
A,
a
contradiction.
Next,
we
observe
that
assertion
(viii)
follows
immediately
from
the
definitions,
in
light
of
the
well-known
fact
that
any
homomorphism
G
→
H
of
group
schemes
over
k,
where
G
is
a
torus
and
H
is
an
abelian
variety,
is
trivial
[cf.,
e.g.,
[BLR],
§10.3,
Theorem
1,
(b),
(c)].
Finally,
we
consider
assertion
(ix).
First,
let
us
observe
that
we
may
assume
without
loss
of
generality
that
k
=
k.
Next,
let
us
observe
that
since
the
image
of
φ
V
contains
0
A
∈
A(k),
it
follows
that
for
n
≥
m,
the
image
I
n
⊆
A(k)
of
ζ
n
contains
the
image
I
m
of
ζ
m
.
Write
F
n
⊆
A
for
the
[reduced
closed
subscheme
given
by
the]
closure
of
I
n
.
Since
the
domain
of
ζ
n
is
irreducible,
it
follows
immediately
that
F
n
is
irreducible.
Thus,
the
ascending
sequence
.
.
.
⊆
F
m
⊆
.
.
.
⊆
F
n
⊆
.
.
.
def
terminates,
i.e.,
we
have
F
n
=
F
m
for
all
n,
m
≥
N
,
for
some
N
;
write
F
=
F
N
.
Since
I
N
is
constructible,
it
follows
that
I
N
contains
a
nonempty
open
subset
U
of
[the
underlying
topological
space
of]
F
;
let
u
∈
U
(k).
Now
let
us
write
I
n
for
the
union
of
the
translates
of
U
by
elements
of
I
n
;
thus,
one
verifies
immediately
that
I
n
is
open
in
F
,
that
I
n
⊆
I
n+N
,
and
that
u
+
I
n
⊆
I
n
.
Since
F
is
noetherian,
it
thus
follows
that
the
ascending
sequence
.
.
.
⊆
I
m
⊆
.
.
.
⊆
I
n
⊆
.
.
.
terminates,
i.e.,
that
for
some
N
>
N
,
we
have
I
n
=
I
m
for
all
n,
m
≥
N
;
write
I
⊆
F
for
the
resulting
open
subscheme.
Thus,
for
n
≥
N
,
u
+
I
⊆
u
+
I
n+N
⊆
I.
On
the
other
hand,
again
since
F
is
noetherian,
it
follows
that
the
ascending
sequence
I
⊆
I
−
u
⊆
I
−
2u
⊆
.
.
.
terminates,
hence
that
u
+
I
=
I.
In
particular,
for
some
N
>
N
,
we
have
I
n
=
I,
for
n
≥
N
.
Next,
let
us
observe
that
for
any
j
∈
I(k),
it
follows
from
the
definition
of
the
I
n
that
j
+
I
⊆
I,
hence
[as
in
the
case
where
j
=
u],
we
have
j
+
I
=
I.
Since
0
A
∈
I,
it
thus
follows
that
I
is
closed
under
the
group
operation
on
A,
as
well
as
taking
inverses
in
A.
Thus,
it
follows
that
I
is
a
subgroup
scheme
of
A,
hence
that
I
is
a
closed
subscheme
of
A
[so
I
=
F
].
But
this
implies,
by
the
definition
of
a
“C-Albanese
morphism”,
the
existence
of
a
homomorphism
A
→
I
whose
composite
with
the
inclusion
I
→
A
is
the
identity
on
A.
Thus,
we
conclude
that
the
inclusion
I
→
A
is
a
surjection,
i.e.,
that
I
=
A,
as
desired.
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
I
71
A
proof
of
the
following
result
may
be
found,
in
essence,
in
[NS]
[albeit
in
somewhat
archaic
language],
as
well
as
in
[FGA],
236,
Théorème
2.1,
(ii)
[albeit
in
somewhat
sketchy
form].
Various
other
approaches
[e.g.,
via
Weil
divisors]
to
this
result
are
discussed
in
[Klei],
Theorem
5.4,
and
the
discussion
following
[Klei],
Theorem
5.4.
Theorem
A.4.
(Properness
of
the
Identity
Component
of
the
Picard
Scheme)
The
identity
component
of
the
Picard
scheme
Pic
0
V
/k
[cf.,
e.g.,
[BLR],
§8.2,
Theorem
3;
[BLR],
§8.4]
associated
to
a
complete
normal
variety
V
over
a
field
k
is
proper.
Proof.
Write
G
for
the
reduced
group
scheme
(Pic
0
V
/k
)
red
over
k.
Then
by
a
well-known
theorem
of
Chevalley
[cf.,
e.g.,
[Con],
for
a
treatment
of
this
result
in
modern
language;
[Bor],
Theorems
10.6,
10.9],
it
follows
that
to
show
that
G
[hence
also
Pic
0
V
/k
]
is
proper,
it
suffices
to
show
that
G
does
not
contain
any
copies
of
the
multiplicative
group
(G
m
)
k
or
the
additive
group
(G
a
)
k
.
On
the
other
hand,
since
(G
m
)
k
,
(G
a
)
k
may
be
thought
of
as
open
subschemes
of
the
affine
line
A
1
k
,
this
follows
immediately
from
Lemma
A.5
below
[i.e.,
by
applying
the
functorial
interpretation
of
Pic
0
V
/k
—
cf.,
e.g.,
[BLR],
§8.1,
Proposition
4].
Lemma
A.5.
(Rational
Families
of
Line
Bundles)
Let
V
be
a
normal
variety
over
k;
U
⊆
A
1
k
a
nonempty
open
subscheme
of
the
affine
line
A
1
k
.
Then
every
line
bundle
L
U
on
V
×
k
U
arises
via
pull-back
from
a
line
bundle
L
k
on
V
.
Proof.
In
the
following,
let
us
regard
A
1
k
as
an
open
subscheme
A
1
k
⊆
P
1
k
of
the
projective
line
[obtained
in
the
standard
way
by
removing
the
point
at
infinity
∞
k
∈
P
1
k
(k)].
First,
let
us
verify
Lemma
A.5
under
the
further
hypothesis
that
V
is
smooth
over
k.
Then
it
follows
immediately
that
V
×
k
P
1
k
is
smooth
over
k,
hence
locally
factorial
[cf.,
e.g.,
[SGA2],
XI,
3.13,
(i)].
Thus,
L
U
extends
to
a
line
def
bundle
L
P
on
P
=
V
×
k
P
1
k
(⊇
V
×
k
A
1
k
⊇
V
×
k
U
).
Moreover,
by
tensoring
with
line
bundles
associated
to
multiples
of
the
divisor
on
P
arising
from
∞
k
,
we
may
assume
that
the
degree
of
L
P
on
the
fibers
of
the
trivial
projective
bundle
f
:
P
→
V
is
zero.
Thus,
the
natural
morphism
f
∗
f
∗
L
P
→
L
P
is
an
isomorphism,
which
exhibits
L
P
,
hence
also
L
U
,
as
a
line
bundle
L
k
pulled
back
from
V
.
Now
we
return
to
the
case
of
an
arbitrary
normal
variety
V
.
As
is
well-known,
V
contains
a
dense
open
subscheme
W
⊆
V
which
is
smooth
over
k
and
such
that
def
the
closed
subscheme
F
=
V
\
W
[where
we
equip
F
with
the
reduced
induced
structure]
is
of
codimension
≥
2
in
V
[cf.,
e.g.,
[SGA2],
XI,
3.11,
applied
to
the
geometric
fiber
of
V
→
Spec(k)].
Thus,
by
the
argument
given
in
the
smooth
case,
def
we
conclude
that
M
U
=
L
U
|
W
×
k
U
arises
from
a
line
bundle
M
k
on
W
.
Next,
let
us
write
ι
k
:
W
→
V
,
ι
U
:
W
×
k
U
→
V
×
k
U
for
the
natural
open
immersions.
Since
U
is
k-flat,
it
follows
immediately
that
we
have
a
natural
isomorphism
∼
((ι
k
)
∗
M
k
)|
V
×
k
U
→
(ι
U
)
∗
M
U
72
SHINICHI
MOCHIZUKI
[arising,
for
instance,
by
computing
the
right-hand
side
by
means
of
an
affine
cov-
ering
of
W
×
k
U
obtained
by
taking
the
product
over
k
with
U
of
an
affine
cov-
ering
of
W
].
On
the
other
hand,
since
V
×
k
U
is
normal
and
F
×
k
U
⊆
V
×
k
U
is
a
closed
subscheme
of
codimension
≥
2,
it
follows
from
the
definition
of
M
U
∼
that
(ι
U
)
∗
M
U
→
L
U
[cf.,
e.g.,
[SGA2],
XI,
3.4;
[SGA2],
XI,
3.11],
i.e.,
that
((ι
k
)
∗
M
k
)|
V
×
k
U
is
a
line
bundle
on
V
×
k
U
.
On
the
other
hand,
since
the
morphism
U
→
Spec(k),
hence
also
the
projection
morphism
V
×
k
U
→
V
,
is
faithfully
flat,
def
we
thus
conclude
that
L
k
=
(ι
k
)
∗
M
k
is
a
line
bundle
on
V
whose
pull-back
to
V
×
k
U
is
isomorphic
to
L
U
,
as
desired.
Proposition
A.6.
(Duals
of
Picard
Varieties
as
Albanese
Varieties)
Let
V
be
a
complete
normal
variety
over
k;
Pic
0
V
/k
the
identity
component
of
def
the
associated
Picard
scheme;
A
the
dual
abelian
variety
to
G
=
(Pic
0
V
/k
)
red
[which
is
an
abelian
variety
by
Theorem
A.4];
v
∈
V
(k).
Then
the
universal
line
bundle
P
V
[cf.,
e.g.,
[BLR],
§8.1,
Proposition
4]
on
V
×
k
G
relative
to
the
rigidification
determined
by
v
[i.e.,
such
that
P
V
|
{v}×G
is
trivial]
determines
[by
the
definition
of
A]
a
morphism
of
pointed
k-varieties
φ
:
V
→
A
such
that
the
pull-back
of
the
Poincaré
bundle
P
A
on
A
×
k
G
via
φ
×
k
G
:
V
×
k
G
→
A
×
k
G
is
isomorphic
to
P
V
[in
a
fashion
compatible
with
the
respective
rigidifications].
Moreover:
(i)
The
morphism
φ
is
a
C
k
ab
-Albanese
morphism.
(ii)
Suppose,
in
the
situation
of
Proposition
A.3,
(ii),
that
W
is
also
complete
and
normal,
and
that
β
V
is
pointed
and
birational.
Then
the
dual
morphism
β
G
:
H
→
G
to
β
A
:
A
→
B
is
a
closed
immersion.
In
particular,
β
A
is
an
isomorphism
if
and
only
if
dim
k
(A)
≤
dim
k
(B).
(iii)
The
morphism
φ
induces
an
injection
H
1
(A,
O
A
)
→
H
1
(V,
O
V
).
∼
→
Δ
A
[where
we
refer
(iv)
The
morphism
φ
induces
an
isomorphism
Δ
ab-t
V
to
§0
for
the
notation
“ab-t”].
Proof.
First,
we
consider
assertion
(i).
Let
ψ
V
:
V
→
C
be
a
morphism
of
pointed
k-varieties,
where
C
∈
C
k
ab
.
Now
by
the
functoriality
of
“Pic
0(−)/k
”,
ψ
V
induces
a
def
morphism
D
=
Pic
0
C/k
→
Pic
0
V
/k
[so
D
is
the
dual
abelian
variety
to
C],
hence
a
morphism
ψ
D
:
D
→
G,
whose
dual
gives
a
morphism
ψ
A
:
A
→
C.
The
fact
that
ψ
V
=
ψ
A
◦φ
:
V
→
C
follows
by
thinking
of
morphisms
as
classifying
morphisms
for
line
bundles
and
considering
the
following
[a
priori,
not
necessarily
commutative]
diagram
of
morphisms
between
varieties
equipped
with
[isomorphism
classes
of]
line
bundles:
(V
×
k
D,
L)
⏐
⏐
ψ
×
D
V
k
−→
id
(V
×
k
D,
L)
⏐
⏐
φ×
D
k
V
×
k
ψ
D
−→
(V
×
k
G,
P
V
)
⏐
⏐
φ×
G
k
(C
×
k
D,
P
C
)
ψ
A
×
k
D
(A
×
k
D,
M)
A×
k
ψ
D
(A
×
k
G,
P
A
)
←−
−→
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
I
73
def
def
—
where
we
write
L
=
(ψ
V
×
k
D)
∗
P
C
;
M
=
(ψ
A
×
k
D)
∗
P
C
∼
=
(A×
k
ψ
D
)
∗
P
A
.
That
is
to
say,
the
desired
commutativity
of
the
left-hand
square
follows
by
computing:
(φ
×
k
D)
∗
(ψ
A
×
k
D)
∗
P
C
∼
=
(φ
×
k
D)
∗
(A
×
k
ψ
D
)
∗
P
A
∼
=
(V
×
k
ψ
D
)
∗
(φ
×
k
G)
∗
P
A
∼
=
(V
×
k
ψ
D
)
∗
P
V
∼
(ψ
V
×
k
D)
∗
P
C
=
—
which
implies
that
ψ
V
=
ψ
A
◦
φ.
Finally,
the
uniqueness
of
such
a
“ψ
A
”
follows
immediately
by
applying
“Pic
0(−)/k
”
to
the
condition
“ψ
V
=
ψ
A
◦
φ
:
V
→
A
→
C”.
This
completes
the
proof
of
assertion
(i).
Next,
we
consider
assertion
(ii).
First,
observe
that
there
exists
a
k-smooth
open
subscheme
U
⊆
W
such
that
W
\
U
has
codimension
≥
2
in
W
[cf.,
e.g.,
[SGA2],
XI,
3.11,
as
it
was
applied
in
the
proof
of
Lemma
A.5],
and,
moreover,
β
V
:
V
→
W
admits
a
section
σ
:
U
→
V
over
U
.
Note,
moreover,
that
if
S
is
any
def
def
local
artinian
finite
k-scheme,
and
we
write
ι
S
:
U
S
=
U
×
k
S
→
W
S
=
W
×
S
k
for
the
natural
inclusion,
then
for
any
line
bundle
L
on
W
S
,
we
have
a
natural
∼
isomorphism
(ι
S
)
∗
(ι
∗
S
L)
→
L
[cf.,
e.g.,
[SGA2],
XI,
3.4;
[SGA2],
XI,
3.11].
Thus,
by
applying
this
natural
isomorphism,
together
with
the
section
σ,
we
conclude
that
the
map
Pic
0
W/k
(S)
→
Pic
0
V
/k
(S)
[induced
by
β
V
]
is
an
injection,
which
implies
that
the
kernel
group
scheme
of
β
G
:
H
→
G
is
trivial,
hence
that
β
G
is
a
closed
immersion,
as
desired.
This
completes
the
proof
of
assertion
(ii).
Next,
we
consider
assertion
(iii).
The
morphism
H
1
(A,
O
A
)
→
H
1
(V,
O
V
)
in
question
may
be
interpreted
as
the
morphism
induced
by
φ
on
tangent
spaces
to
the
Picard
scheme,
i.e.,
as
the
morphism
G(k[]/(
2
))
=
Pic
0
A/k
(k[]/(
2
))
→
Pic
0
V
/k
(k[]/(
2
))
[cf.,
e.g.,
[BLR],
§8.4,
Theorem
1,
(a)].
But,
by
the
definition
of
G,
this
morphism
arises
from
the
natural
closed
immersion
G
→
Pic
0
V
/k
,
hence
is
an
injection,
as
desired.
Finally,
we
consider
assertion
(iv).
The
surjectivity
portion
of
assertion
(iv)
follows
immediately
from
Proposition
A.3,
(vii).
To
verify
the
fact
that
the
sur-
jection
Δ
ab-t
Δ
A
is
an
isomorphism,
we
reason
as
follows:
First,
we
recall
that
V
if
n
≥
1
is
an
integer
invertible
in
k,
then
a
line
bundle
L
on
V
such
that
L
⊗n
is
trivial
may
be
interpreted
[via
the
Kummer
exact
sequence
in
étale
cohomol-
ogy]
as
a
continuous
homomorphism
Δ
V
→
(Z/nZ)(1)
[where
the
“(1)”
denotes
a
“Tate
twist”].
On
the
other
hand,
by
[BLR],
§8.4,
Theorem
7,
there
exists
an
integer
m
≥
1
such
that
for
every
integer
n
≥
1,
the
cokernel
of
the
inclusion
n
G(k)
→
n
Pic
V
/k
(k)
[where
the
“
n
”
preceding
an
abelian
group
denotes
the
kernel
of
multiplication
by
n]
is
annihilated
by
m.
In
light
of
the
functorial
interpretation
of
the
inclusion
G
→
Pic
0
V
/k
⊆
Pic
V
/k
,
this
implies
that
the
cokernel
of
the
homo-
morphism
Hom(Δ
A
,
Q/Z)
→
Hom(Δ
V
,
Q/Z)
is
annihilated
by
m.
But,
by
applying
Hom(−,
Q/Z),
this
implies
that
the
induced
homomorphism
Δ
ab
V
→
Δ
A
has
finite
kernel,
hence
[in
light
of
the
surjectivity
already
verified]
induces
an
isomorphism
upon
passing
to
“ab-t”.
74
SHINICHI
MOCHIZUKI
Remark
A.6.1.
The
content
of
Proposition
A.6,
(i),
is
discussed
in
[FGA],
236,
Théorème
3.3,
(iii).
Remark
A.6.2.
Suppose
that
we
are
in
the
situation
of
Proposition
A.6,
(ii).
Then
it
is
not
necessarily
the
case
that
the
induced
morphism
β
A
is
an
isomorphism.
This
phenomenon
already
appears
in
the
work
of
Chevalley
—
cf.
[Chev];
the
discussion
of
[Klei],
p.
248;
Example
A.7
below.
Example
A.7.
Albanese
Varieties
and
Resolution
of
Singularities.
For
simplicity,
suppose
that
k
=
k.
Write
P
2
k
=
Proj(k[x
1
,
x
2
,
x
3
])
[i.e.,
where
we
consider
k[x
1
,
x
2
,
x
3
]
as
a
graded
ring,
in
which
x
1
,
x
2
,
x
3
are
of
degree
1].
Let
f
∈
k[x
1
,
x
2
,
x
3
]
be
a
homogeneous
polynomial
that
defines
a
smooth
plane
curve
X
⊆
P
2
k
of
genus
≥
1.
Thus,
any
x
∈
X(k)
determines
an
embedding
X
→
J,
where
def
J
is
the
Jacobian
variety
of
X.
Set
Y
=
Spec(k[x
1
,
x
2
,
x
3
]/(f
));
write
y
∈
Y
(k)
def
for
the
origin,
U
Y
=
Y
\
{y}.
Thus,
we
have
a
natural
morphism
Y
⊇
U
Y
→
X;
U
Y
→
X
is
a
G
m
-torsor
over
X.
In
particular,
U
Y
is
k-smooth.
Thus,
since
Y
is
clearly
a
local
complete
intersection
[hence,
in
particular,
Cohen-Macaulay],
it
follows
from
Serre’s
criterion
of
normality
[cf.,
e.g.,
[SGA2],
XI,
3.11]
that
Y
is
normal.
Let
Z
→
Y
be
the
blow-up
of
Y
at
the
origin
y.
Thus,
we
obtain
an
def
∼
isomorphism
U
Z
=
Z
×
Y
U
Y
→
U
Y
.
Moreover,
one
verifies
immediately
that
the
∼
morphism
U
Z
→
U
Y
→
X
extends
to
a
morphism
Z
→
X
which
has
the
structure
def
∼
of
an
A
1
-bundle,
in
which
E
=
Z
×
Y
{y}
⊆
Z
forms
a
“zero
section”
[so
E
→
X].
Thus,
Z
admits
a
natural
compactification
Z
→
Z
∗
to
a
P
1
-bundle
Z
∗
→
X.
∼
Moreover,
by
gluing
Z
∗
\
E
to
Y
along
Z
\
E
=
U
Z
→
U
Y
⊆
Y
,
we
obtain
a
compactification
Y
→
Y
∗
such
that
the
blow-up
morphism
extends
to
a
morphism
Z
∗
→
Y
∗
[which
may
be
thought
of
as
the
blow-up
of
Y
∗
at
y
∈
Y
(k)
⊆
Y
∗
(k)].
On
the
other
hand,
note
that
the
composite
Z
∗
→
X
→
J
determines
a
closed
∼
∼
immersion
Z
∗
⊇
E
→
X
→
J.
Thus,
the
restriction
U
Y
→
U
Z
→
J
of
this
∼
morphism
Z
∗
→
J
to
U
Y
→
U
Z
does
not
extend
to
Y
or
Y
∗
.
In
particular,
it
follows
that
if
we
write
Y
∗
→
A
Y
,
Z
∗
→
A
Z
for
the
C
k
ab
-Albanese
varieties
of
Proposition
A.6,
(i),
then
the
surjection
A
Z
A
Y
induced
by
Z
∗
→
Y
∗
[cf.
Proposition
A.6,
(ii)]
is
not
an
isomorphism.
Proposition
A.8.
(Albanese
Varieties
of
Complements
of
Divisors
with
Normal
Crossings)
Let
Z
be
a
smooth
projective
variety
over
k;
D
⊆
Z
a
def
divisor
with
normal
crossings;
Y
=
Z
\
D
⊆
Z;
y
∈
Y
(k);
D
=
r
D
n
n=1
[for
some
integer
r
≥
1]
the
decomposition
of
D
into
irreducible
components;
M
the
free
Z-module
[of
rank
r]
of
divisors
supported
on
D;
P
⊆
M
the
submodule
of
divisors
that
determine
a
line
bundle
∈
Pic
0
Z/k
(k).
Then:
(i)
(Y,
y)
admits
an
Albanese
morphism
Y
→
A
Y
.
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
I
75
(ii)
Suppose
that
each
of
the
D
n
is
geometrically
irreducible.
Then
the
A
Y
of
(i)
may
be
taken
to
be
an
extension
of
the
abelian
variety
A
Z
given
by
the
def
dual
to
G
Z
=
(Pic
0
Z/k
)
red
[cf.
Propositions
A.3,
(viii);
A.6,
(i)]
by
a
torus
whose
character
group
is
naturally
isomorphic
to
P
.
∼
(iii)
The
morphism
Y
→
A
Y
of
(i)
induces
an
isomorphism
Δ
ab-t
→
Δ
A
Y
.
Y
Proof.
By
étale
descent
[with
respect
to
finite
extensions
of
k],
it
follows
immedi-
ately
that
to
verify
assertion
(i),
it
suffices
to
verify
assertion
(ii).
Next,
we
consider
assertion
(ii).
Again,
by
étale
descent,
we
may
assume
without
loss
of
generality
that
k
=
k.
Note
that
the
tautological
homomorphism
P
→
G
Z
(k)
determines
an
extension
0
→
T
Y
→
A
Y
→
A
Z
→
0
of
A
Z
by
a
split
torus
T
Y
with
character
group
P
.
Now
the
fact
that
A
Y
serves
as
an
Albanese
variety
for
Y
is
essentially
a
tautology:
Indeed,
since
any
pointed
morphism
from
Y
to
an
abelian
variety
C
extends
[cf.
Lemma
A.2]
to
a
pointed
morphism
Z
→
C,
and,
moreover,
we
already
know
that
A
Z
is
a
C
k
ab
-Albanese
variety
for
Z
[cf.
Proposition
A.6,
(i)],
it
follows
that
it
suffices
to
consider
pointed
morphisms
Y
→
B,
where
B
is
an
extension
of
A
Z
by
a
[split]
torus,
and
the
composite
morphism
Y
→
B
A
Z
coincides
with
the
morphism
that
exhibits
A
Z
as
a
C
k
ab
-Albanese
variety
for
Y
.
In
fact,
for
simplicity,
we
may
even
assume
that
this
torus
is
simply
(G
m
)
k
.
Thus,
it
suffices
to
consider
pointed
morphisms
Y
→
B,
where
B
is
an
extension
of
A
Z
by
(G
m
)
k
,
determined
by
some
extension
class
κ
B
∈
G
Z
(k),
and
the
composite
morphism
Y
→
B
A
Z
coincides
with
the
morphism
that
exhibits
A
Z
as
a
C
k
ab
-Albanese
variety
for
Y
.
Then
the
datum
of
such
a
morphism
Y
→
B
corresponds
precisely
to
an
invertible
section
of
the
restriction
to
Y
of
the
line
bundle
L
on
Z
given
by
pulling
back
the
G
m
-torsor
B
→
A
Z
via
the
Albanese
morphism
Z
→
A
Z
.
Note
that
such
an
invertible
section
∼
of
L|
Y
may
be
thought
of
as
the
datum
of
an
isomorphism
O
Z
(E)
→
L
for
some
divisor
E
supported
on
D.
That
is
to
say,
since
the
isomorphism
class
of
L
is
precisely
the
class
determined
by
the
element
κ
B
∈
G
Z
(k)
⊆
Pic
Z/k
(k),
it
thus
follows
that
E
∈
P
,
and
that
κ
B
is
the
image
of
E
∈
P
in
Pic
0
Z/k
(k)
=
G
Z
(k).
Thus,
in
summary,
the
datum
of
a
pointed
morphism
Y
→
B,
where
B
is
an
extension
of
A
Z
by
a
[split]
torus,
and
the
composite
morphism
Y
→
B
A
Z
coincides
with
the
morphism
that
exhibits
A
Z
as
a
C
k
ab
-Albanese
variety
for
Y
,
is
equivalent
[in
a
functorial
way]
to
the
datum
of
a
homomorphism
A
Y
→
B
lying
over
the
identity
morphism
of
A
Z
.
In
particular,
the
identity
morphism
A
Y
→
A
Y
determines
a
morphism
Y
→
A
Y
.
This
completes
the
proof
of
assertion
(ii).
Finally,
we
consider
assertion
(iii).
We
may
assume
without
loss
of
generality
that
k
=
k
[cf.
Proposition
A.3,
(i)].
Let
F
⊆
D
be
a
closed
subscheme
of
def
def
codimension
≥
1
in
D
such
that
Z
=
Z
\
F
⊆
Z,
D
=
D
\
F
⊆
D
are
k-smooth.
Then
one
has
the
associated
Gysin
sequence
in
étale
cohomology
1
1
2
(Z
,
Z
l
(1))
→
H
ét
(Y,
Z
l
(1))
→
M
⊗
Z
l
→
H
ét
(Z
,
Z
l
(1))
0
→
H
ét
for
l
∈
Σ
k
[cf.
[Milne],
p.
244,
Remark
5.4,
(b)].
Moreover,
we
have
natural
isomor-
∼
j
j
(Z
,
Z
l
(1))
→
H
ét
(Z,
Z
l
(1)),
for
j
=
1,
2.
[Indeed,
by
applying
noetherian
phisms
H
ét
76
SHINICHI
MOCHIZUKI
induction,
it
suffices
to
verify
these
isomorphisms
in
the
case
where
F
is
k-smooth,
in
which
case
these
isomorphisms
follow
from
[Milne],
p.
244,
Remark
5.4,
(b).]
∼
2
2
Note,
moreover,
that
the
morphism
M
⊗
Z
l
→
H
ét
(Z
,
Z
l
(1))
→
H
ét
(Z,
Z
l
(1))
is
precisely
the
“fundamental
class
map”,
hence
factors
through
the
natural
inclusion
2
Pic
Z/k
(k)
∧
→
H
ét
(Z,
Z
l
(1))
[where
the
“∧”
denotes
the
pro-l
completion]
arising
from
the
Kummer
exact
se-
quence
on
Z.
On
the
other
hand,
since
Pic
0
Z/k
(k)
is
l-divisible,
and
the
quotient
Pic
Z/k
(k)/Pic
0
Z/k
(k)
is
finitely
generated
[cf.
[BLR],
§8.4,
Theorem
7],
it
follows
that
we
have
an
isomorphism
∼
(Pic
Z/k
(k)/Pic
0
Z/k
(k))
⊗
Z
l
→
Pic
Z/k
(k)
∧
2
—
i.e.,
that
the
kernel
of
the
morphism
M
⊗Z
l
→
H
ét
(Z
,
Z
l
(1))
is
precisely
P
⊗Z
l
.
∼
→
Δ
A
Z
of
Proposition
A.6,
(iv),
implies
[in
In
particular,
the
isomorphism
Δ
ab-t
Z
1
light
of
the
above
exact
sequence]
that
H
ét
(Y,
Z
l
(1))
[i.e.,
Hom(Δ
ab-t
Y
,
Z
l
(1))],
hence
ab-t
also
Δ
Y
⊗Z
l
,
is
a
free
Z
l
-module
of
the
same
rank
as
Δ
A
Y
⊗Z
l
.
Thus,
we
conclude
that
the
surjection
Δ
ab-t
Δ
A
Y
of
Proposition
A.3,
(vii),
is
an
isomorphism,
as
Y
desired.
Remark
A.8.1.
A
sharper
version
[in
the
sense
that
it
includes
a
computation
of
the
torsion
subgroup
of
Δ
ab
Y
]
of
Proposition
A.8,
(iii),
is
given
in
[SS],
Proposition
4.2.
The
discussion
of
[SS]
involves
the
point
of
view
of
1-motives.
On
the
other
hand,
such
a
sharper
version
may
also
be
obtained
directly
from
the
Gysin
sequence
argument
of
the
above
proof
of
Proposition
A.8,
(iii),
by
working
with
torsion
coefficients.
The
following
result
is
elementary
and
well-known.
Lemma
A.9.
(Descending
Chains
of
Subgroup
Schemes)
Let
G
be
a
[not
necessarily
reduced]
commutative
group
scheme
of
finite
type
over
k;
.
.
.
⊆
G
n
⊆
.
.
.
⊆
G
1
⊆
G
0
=
G
a
descending
chain
of
[not
necessarily
reduced!]
subgroup
schemes
of
G,
indexed
by
the
nonnegative
integers.
Then
there
exists
an
integer
N
such
that
G
n
=
G
m
for
all
n,
m
≥
N
.
Proof.
First,
let
us
consider
the
case
where
all
of
the
G
n
,
for
n
≥
0,
are
reduced
and
connected.
Then
since
all
of
the
G
n
are
closed
integral
subschemes
of
G,
it
follows
immediately
that
if
we
take
any
integer
N
such
that
dim
k
(G
n
)
=
dim
k
(G
m
)
for
all
n,
m
≥
N
,
then
G
n
=
G
m
for
all
n,
m
≥
N
.
Now
we
return
to
the
general
case.
By
what
we
have
done
so
far,
we
may
assume
without
loss
of
generality
that
(G
0
)
red
=
(G
n
)
red
for
all
n
≥
0.
Thus,
by
forming
the
quotient
by
(G
0
)
red
,
we
may
assume
that
all
of
the
G
n
are
finite
over
Spec(k).
Then
Lemma
A.9
follows
immediately.
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
I
77
Before
proceeding,
we
recall
the
following
result
of
de
Jong.
Lemma
A.10.
(Equivariant
Alterations)
Suppose
that
k
=
k;
let
V
be
a
variety
over
k.
Then
there
exists
a
smooth
projective
variety
Z
over
k,
a
finite
group
Γ
of
automorphisms
of
Z
over
k,
a
divisor
with
normal
crossings
D
⊆
Z
stabilized
by
Γ,
and
a
Γ-equivariant
[relative
to
the
trivial
action
of
Γ
on
V
]
surjective,
proper,
generically
quasi-finite
morphism
def
Y
=
Z
\
D
→
V
such
that
if
we
write
k(Z),
k(V
)
for
the
respective
function
fields
of
Z,
V
,
then
the
subfield
of
Γ-invariants
k(Z)
Γ
⊆
k(Z)
forms
a
purely
inseparable
extension
of
k(V
).
Proof.
This
is
the
content
of
[deJong],
Theorem
7.3.
We
are
now
ready
to
prove
the
main
result
of
the
present
Appendix,
the
first
portion
of
which
[i.e.,
Corollary
A.11,
(i)]
is
due
to
Serre
[cf.
[Serre1]].
Corollary
A.11.
(Albanese
Varieties
of
Arbitrary
Varieties)
(i)
Every
pointed
variety
(V,
v)
over
k
admits
an
Albanese
morphism
V
→
A.
(ii)
Let
φ
:
V
→
A
be
an
Albanese
morphism,
where
(V,
v)
is
a
k-toric
∼
pointed
variety.
Then
φ
induces
an
isomorphism
Δ
ab-t
→
Δ
A
.
V
Proof.
First,
we
consider
assertion
(i).
By
applying
étale
descent,
we
may
assume
without
loss
of
generality
that
k
=
k.
Let
Z
⊇
Y
→
V
be
as
in
Lemma
A.10,
y
∈
Y
(k)
a
point
that
maps
to
v
∈
V
(k)
[where
we
observe
that,
as
is
easily
verified,
the
existence
of
an
Albanese
morphism
as
desired
is
independent
of
the
choice
of
v].
Then
by
Proposition
A.8,
(i),
it
follows
that
Y
admits
an
Albanese
morphism
Y
→
B.
Thus,
every
pointed
morphism
ν
:
V
→
C,
where
C
∈
C
k
s-ab
,
determines,
by
restriction
to
Y
,
a
homomorphism
B
→
C,
whose
kernel
is
a
subgroup
scheme
H
ν
⊆
B.
In
particular,
the
collection
of
such
pointed
morphisms
ν
:
V
→
C
determines
a
projective
system
of
subgroup
schemes
H
ν
⊆
B
which
is
filtered
[a
fact
that
is
easily
verified
by
considering
product
morphisms
V
→
C
1
×
k
C
2
of
pointed
morphisms
ν
1
:
V
→
C
1
,
ν
2
:
V
→
C
2
].
Moreover,
by
Lemma
A.9,
this
projective
system
admits
a
cofinal
subsystem
which
is
constant,
i.e.,
given
by
a
single
subgroup
scheme
H
⊆
B.
Now
it
is
a
tautology
that
the
composite
morphism
Y
→
B
B/H
factors
uniquely
[where
we
observe
that
uniqueness
follows
from
the
fact
that
Y
→
V
is
dominant]
through
a
morphism
V
→
B/H
which
serves
as
an
Albanese
morphism
for
V
.
Next,
we
consider
assertion
(ii).
First,
let
us
observe
that,
by
Proposition
A.3,
(i),
we
may
assume
without
loss
of
generality
that
k
=
k.
Next,
let
Z
⊇
Y
→
V
,
Γ
be
as
in
Lemma
A.10;
write
Y
→
V
→
V
for
the
factorization
through
the
normalization
V
→
V
of
V
in
the
purely
inseparable
extension
k(Z)
Γ
of
k(V
).
Let
78
SHINICHI
MOCHIZUKI
φ
:
V
→
A
be
an
Albanese
morphism
[which
exists
by
assertion
(i)].
Since
V
is
normal,
it
follows
immediately
that
V
→
V
is
a
sub-Frobenius
morphism.
Thus,
by
Proposition
A.3,
(iv)
[cf.
also
Remark
A.1.1],
it
follows
that
V
→
V
induces
∼
∼
ab-t
isomorphisms
Δ
ab-t
V
→
Δ
V
,
Δ
A
→
Δ
A
.
In
particular,
to
complete
the
proof
of
∼
assertion
(ii),
it
suffices
to
verify
that
φ
induces
an
isomorphism
Δ
ab-t
V
→
Δ
A
.
Next,
let
Y
→
B
be
an
Albanese
morphism
for
Y
[cf.
Proposition
A.8,
(i)].
Then,
by
Proposition
A.3,
(ii),
the
action
of
Γ
on
Y
extends
to
a
compatible
action
of
Γ
on
B
by
k-trans-homomorphisms.
This
action
of
Γ
on
B
may
be
thought
of
as
the
combination
of
an
action
of
Γ
on
the
group
scheme
B
[i.e.,
via
group
scheme
automorphisms],
together
with
a
twisted
homomorphism
χ
:
Γ
→
B(k)
[where
Γ
acts
on
B(k)
via
the
group
scheme
action
of
Γ
on
B].
Write
B
C
for
the
quotient
semi-abelian
scheme
of
B
by
the
group
scheme
action
Γ,
i.e.,
the
quotient
of
B
by
the
subgroup
scheme
generated
by
the
images
of
the
group
scheme
endomorphisms
(1
−
γ)
:
B
→
B,
for
γ
∈
Γ.
Thus,
χ
determines
a
homomorphism
χ
:
Γ
→
C
(k);
write
C
→
C
for
the
quotient
semi-abelian
scheme
of
C
by
the
finite
subgroup
scheme
of
C
determined
by
the
image
of
χ
.
Note
that
every
trans-homomorphism
of
semi-abelian
schemes
B
→
D
which
is
Γ-equivariant
with
respect
to
the
trivial
action
of
Γ
on
D
and
the
trans-homomorphism
action
of
Γ
of
B
factors
uniquely
through
B
C.
Now
I
claim
that
the
composite
Y
→
B
C
factors
uniquely
through
V
.
Indeed,
this
is
clear
generically;
write
ξ
:
η
V
→
C
for
the
resulting
morphism.
Here,
we
use
the
notation
“η
(−)
”
to
denote
the
spectrum
of
the
function
field
of
“(−)”.
Since
the
morphism
V
→
V
is
a
sub-Frobenius
morphism,
it
thus
follows
n
that
for
some
integer
n
≥
1,
the
composite
η
V
→
C
→
C
F
of
ξ
with
Φ
nC
factors
n
through
the
natural
morphism
η
V
→
η
V
,
thus
yielding
a
morphism
ξ
:
η
V
→
C
F
.
Now
since
V
is
normal,
it
follows
from
the
properness
of
Y
→
V
that
ξ
extends
uniquely
to
points
of
height
1
of
V
;
thus,
since
V
is
k-toric,
it
follows
from
Lemma
A.2
that
ξ
extends
uniquely
to
the
entire
scheme
V
.
Finally,
by
the
definition
of
V
→
V
as
a
normalization
morphism,
it
follows
that
from
the
fact
that
Φ
nC
is
finite
and
surjective
that
ξ
extends
uniquely
to
the
entire
scheme
V
.
This
completes
the
proof
of
the
claim.
Next,
let
us
observe
that
it
is
a
tautology
that
the
morphism
V
→
C
resulting
from
the
above
claim
is
an
Albanese
morphism
for
V
.
In
particular,
we
may
assume
without
loss
of
generality
that
φ
:
V
→
A
is
V
→
C.
Next,
let
us
observe
that
it
follows
immediately
from
the
description
of
finite
étale
coverings
of
semi-abelian
schemes
reviewed
at
the
beginning
of
the
present
Appendix
that
the
functor
“(−)
→
Δ
(−)
”
transforms
exact
sequences
of
semi-abelian
schemes
into
exact
sequences
of
profinite
groups.
Thus,
if
follows
immediately
from
the
construction
of
C
(=
A
)
from
B
that,
for
l
∈
Σ
k
,
the
surjection
Δ
B
⊗Q
l
Δ
A
⊗Q
l
induces
an
isomorphism
∼
(Δ
B
⊗
Q
l
)/Γ
→
Δ
A
⊗
Q
l
[where
the
“/Γ”
denotes
the
maximal
quotient
on
which
Γ
acts
trivially].
On
the
other
hand,
by
Proposition
A.8,
(iii),
it
follows
that
we
have
a
natural
∼
→
Δ
B
,
hence,
in
particular,
a
natural
isomorphism
isomorphism
Δ
ab-t
Y
∼
∼
⊗
Q
l
)/Γ
→
(Δ
B
⊗
Q
l
)/Γ
→
Δ
A
⊗
Q
l
(Δ
ab-t
Y
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
I
79
for
l
∈
Σ
k
.
Moreover,
since
the
morphism
Y
→
V
is
dominant,
it
induces
an
open
homomorphism
Δ
Y
→
Δ
V
,
hence
a
surjection
Δ
ab-t
⊗
Q
l
Δ
ab-t
Y
V
⊗
Q
l
which
is
Γ-equivariant
[with
respect
to
the
trivial
action
of
Γ
on
Δ
ab-t
⊗
Q
l
].
In
particular,
V
∼
ab-t
we
obtain
that
the
natural
isomorphism
(Δ
Y
⊗
Q
l
)/Γ
→
Δ
A
⊗
Q
l
factors
as
the
composite
of
surjections
(Δ
ab-t
⊗
Q
l
)/Γ
Δ
ab-t
Y
V
⊗
Q
l
Δ
A
⊗
Q
l
[cf.
Proposition
A.3,
(vii)].
Thus,
we
conclude
that
these
surjections
are
isomor-
phisms,
hence
that
the
surjection
Δ
ab-t
V
Δ
A
induced
by
φ
[cf.
Proposition
A.3,
(vii)],
is
an
isomorphism,
as
desired.
Remark
A.11.1.
In
fact,
given
any
variety
V
over
k,
one
may
construct
an
“Albanese
morphism”
V
→
A,
where
A
is
a
torsor
over
a
semi-abelian
variety
over
k,
by
passing
to
a
finite
[separable]
extension
k
of
k
such
that
V
(k
)
=
∅,
applying
Corollary
A.11,
(i),
over
k
,
and
then
descending
back
to
k.
This
morphism
V
→
A
will
then
satisfy
the
universal
property
for
morphisms
V
→
A
to
torsors
A
over
semi-abelian
varieties
over
k
[i.e.,
every
such
morphism
V
→
A
admits
a
unique
factorization
V
→
A
→
A
,
where
the
morphism
A
→
A
is
a
k-morphism
that
base-
changes
to
a
trans-homomorphism
over
k].
In
the
present
Appendix,
however,
we
always
assumed
the
existence
of
rational
points
in
order
to
simplify
the
discussion.
Remark
A.11.2.
One
may
further
generalize
Remark
A.11.1,
as
follows.
If
V
is
a
generically
scheme-like
[cf.
§0]
geometrically
integral
separated
algebraic
stack
of
finite
type
over
k
that
is
obtained
by
forming
the
quotient,
in
the
sense
of
stacks,
of
some
variety
W
over
k
by
the
action
of
a
finite
group
of
automorphisms
Γ
⊆
Aut(W
),
then,
by
applying
Remark
A.11.1
to
W
to
obtain
an
Albanese
morphism
W
→
B
for
W
,
one
may
construct
an
“Albanese
morphism”
V
→
A
for
V
[i.e.,
which
satisfies
the
universal
property
described
in
Remark
A.11.1]
by
forming
the
quotient
B
→
A
of
B
as
in
the
proof
of
Corollary
A.11,
(ii):
That
is
to
say,
after
reducing,
via
étale
descent,
to
the
case
k
=
k,
the
action
of
Γ
on
W
induces
an
action
of
Γ
by
k-trans-homomorphisms
on
B,
hence
an
action
of
Γ
by
group
scheme
automorphisms
on
B,
together
with
a
twisted
homomorphism
χ
:
Γ
→
B(k).
Then
we
take
B
A
to
be
the
quotient
by
the
images
of
the
group
scheme
endomorphisms
[arising
from
the
group
scheme
action
of
Γ
on
B]
(1
−
γ)
:
B
→
B,
for
γ
∈
Γ,
and
A
A
to
be
the
quotient
by
the
image
of
the
homomorphism
χ
:
Γ
→
A
(k)
determined
by
χ.
If,
moreover,
V
[i.e.,
W
]
is
k-toric,
then
just
as
in
the
proof
of
Corollary
A.11,
(ii),
we
obtain
a
natural
isomorphism
∼
Δ
ab-t
→
Δ
A
V
[where
we
use
the
notation
“Δ
(−)
”
to
denote
the
evident
stack-theoretic
general-
ization
of
this
notation
for
varieties].
The
content
of
more
classical
works
[cf.,
e.g.,
[NS],
[Chev]]
written
from
the
point
of
view
of
birational
geometry
may
be
recovered
via
the
following
result.
80
SHINICHI
MOCHIZUKI
Corollary
A.12.
(Albanese
Varieties
and
Birational
Geometry)
(i)
Let
β
V
:
V
→
V
be
a
proper
birational
morphism
of
normal
varieties
def
∼
over
k
which
restricts
to
an
isomorphism
β
U
:
U
=
V
×
V
U
→
U
over
some
nonempty
open
subscheme
U
⊆
V
;
β
A
:
A
→
A
the
induced
morphism
on
Al-
banese
varieties
[cf.
Corollary
A.11,
(i)];
W
⊆
V
a
k-toric
open
subscheme.
∼
Then
the
composite
morphism
U
W
→
U
→
U
→
V
→
A
extends
uniquely
to
a
morphism
W
→
A
which
induces
a
surjection
Δ
W
Δ
A
.
(ii)
Let
.
.
.
→
V
n
→
.
.
.
→
V
1
→
V
0
=
V
be
a
sequence
[indexed
by
the
nonnegative
integers]
of
birational
morphisms
of
complete
normal
varieties
over
k.
Then
there
exists
an
integer
N
such
that
for
all
n,
m
≥
N
,
where
n
≥
m,
the
induced
morphism
on
Albanese
varieties
A
n
→
A
m
is
an
isomorphism.
If
V
is
k-toric,
then
one
may
take
N
=
0.
Proof.
First,
we
consider
assertion
(i).
We
may
assume
without
loss
of
generality
that
U
⊆
W
.
Then
since
V
→
V
is
proper,
and
W
is
normal,
it
follows
that
∼
the
morphism
U
→
U
→
V
extends
uniquely
to
an
open
subset
W
\
F
⊆
W
,
where
F
is
a
closed
subscheme
of
codimension
≥
2
in
W
.
Thus,
the
fact
that
the
resulting
morphism
W
\
F
→
V
→
A
extends
uniquely
to
W
follows
immediately
from
Lemma
A.2.
To
verify
the
surjectivity
of
Δ
W
→
Δ
A
,
it
suffices
to
verify
the
surjectivity
of
Δ
U
→
Δ
A
,
i.e.,
of
Δ
U
→
Δ
V
→
Δ
A
.
On
the
other
hand,
this
follows
from
the
surjectivity
of
Δ
V
→
Δ
A
[cf.
Proposition
A.3,
(vii)],
together
with
the
surjectivity
of
Δ
U
→
Δ
V
[cf.
the
fact
that
U
⊆
V
is
a
nonempty
open
subscheme
of
the
normal
variety
V
].
Next,
we
consider
assertion
(ii).
By
Proposition
A.6,
(i),
(ii)
[cf.
also
Propo-
sition
A.3,
(viii),
(ix);
Corollary
A.11,
(i)],
each
induced
morphism
on
Albanese
varieties
A
n
→
A
m
,
for
n
≥
m,
is
a
surjection
of
abelian
varieties
which
is
an
isomorphism
if
and
only
if
dim
k
(A
n
)
≤
dim
k
(A
m
).
On
the
other
hand,
if
W
⊆
V
is
any
nonempty
k-toric
[e.g.,
k-smooth]
open
subscheme,
whose
Albanese
mor-
phism
[cf.
Corollary
A.11,
(i)]
we
denote
by
W
→
A
W
,
then
assertion
(i)
yields
a
morphism
W
→
A
n
that
induces
a
surjection
Δ
W
Δ
A
n
,
hence,
in
particular,
a
morphism
A
W
→
A
n
that
induces
a
surjection
Δ
A
W
Δ
A
n
.
But
this
implies
that
dim
k
(A
n
)
≤
dim
k
(A
W
),
hence
that
for
some
integer
N
,
dim
k
(A
n
)
=
dim
k
(A
m
),
for
all
n,
m
≥
N
.
In
particular,
if
W
=
V
,
then
dim
k
(A
n
)
≤
dim
k
(A
0
),
for
all
n
≥
0.
TOPICS
IN
ABSOLUTE
ANABELIAN
GEOMETRY
I
81
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